LIMIT PROPERTIES – Examples of using the 8 properties

I’ve already talked a bit about limits and one-sided limits and how to evaluate them, especially using the graph of the functions. But most limits that you need to evaluate won’t come with a graph and may be challenging to graph. In cases like these, you will want to try applying the 8 basic limit properties.

Using the limit properties is the simplest way to evaluate limits. Therefore, applying limit properties should be a good starting place for most limits. These properties can be applied to two-sided and one-sided limits.

First I will go ahead and list the 8 limit properties then I will show you a handful of examples that show how to apply these limits. These are the same 8 limit properties that are listed on my calculus 1 study guide. If you haven’t already, click here to download my calculus 1 study guide so you can have these limit properties handy as you work through evaluating limits with them.

What are the 8 limit properties?

\mathbf{1. \ \ \lim\limits_{x \to a} c = c}

Taking the limit of a constant just results in that constant.

\mathbf{2. \ \ \lim\limits_{x \to a} x = a}

The limit of the variable alone will go toward the value that the variable is approaching as given in the limit. This is a result of the fact that y=x is a continuous function.

\mathbf{3. \ \ \lim\limits_{x \to a} \Big( cf(x) \Big) = c \cdot \lim\limits_{x \to a} f(x)}

Having a constant being multiplied by the entire function within the limit can be pulled out of the limit. This will allow you to evaluate the simpler function, then multiply the result by that constant after evaluating a slightly simpler limit.

\mathbf{4. \ \ \lim\limits_{x \to a} \Big( f(x) \pm g(x) \Big) = \lim\limits_{x \to a} f(x) \pm \lim\limits_{x \to a} g(x)}

The limit of a sum or difference can instead be written as the sum or difference of their individual limits.

\mathbf{5. \ \ \lim\limits_{x \to a} \Big( f(x) \cdot g(x) \Big) = \lim\limits_{x \to a} f(x) \cdot \lim\limits_{x \to a} g(x)}

The limit of a product can instead be written as the product of their individual limits.

\mathbf{6. \ \ \lim\limits_{x \to a} \Big( \frac{f(x)}{g(x)} \Big) = \frac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)}, \ \ if \ \lim\limits_{x \to a} g(x) \neq 0}

Taking the limit of a quotient can be rewritten as the quotient of those two limits. Just make sure that the limit of the denominator isn’t zero. If it is, then this will result in dividing by zero, which you can’t do.

\mathbf{7. \ \ \lim\limits_{x \to a} \Big( f(x) \Big)^n = \Big( \lim\limits_{x \to a} f(x) \Big)^n}

Taking the limit of some function raised to a constant power can be rewritten to evaluate the limit of the inner function then raise the result to that constant power.

\mathbf{8. \ \ \lim\limits_{x \to a} \Big( \sqrt[\leftroot{-3}\uproot{3}n]{f(x)} \Big) = \sqrt[\leftroot{-3}\uproot{3}n]{\lim\limits_{x \to a} f(x)}}

Similar to the last property, but the same can be done with a function that is within a root. This can be applied to any constant root (eg. square root, cube root, etc.)

How can these limit properties be applied to evaluate limits?

These 8 properties of limits can be used to simplify limits and break them down into smaller pieces. Each of these smaller pieces would be easier to deal with. Then once you evaluate these smaller, simpler limits you can put them all together.

We will go ahead and show how to apply these limit properties with some examples. To the right of each step in parenthesis, I will put a number corresponding to the property from above that was used to get to that step from the previous one. If multiple properties were applied at the same time I will list all properties used in that step in the parenthesis.

Example 1

$$\lim_{x \to 5} 6x^4 – 2x + 7$$ $$= \ \lim_{x \to 5} 6x^4 – \lim_{x \to 5} 2x + \lim_{x \to 5} 7 \ \ \ \ (4)$$ $$= \ 6 \lim_{x \to 5} x^4 – 2 \lim_{x \to 5} x + \lim_{x \to 5} 7 \ \ \ \ (3)$$ $$= \ 6 \Big( \lim_{x \to 5} x \Big)^4 – 2 \lim_{x \to 5} x + \lim_{x \to 5} 7 \ \ \ \ (7)$$ $$= \ 6 (5)^4 – 2(5) + 7 \ \ \ \ (1, \ 2)$$ $$= \ 3,747$$

Example 2

$$\lim_{x \to 2} \Big( (x+2) \sqrt[\leftroot{-1}\uproot{3}3]{x^2 + 7x} \Big)$$ $$= \ \lim_{x \to 2}(x+2) \cdot \lim_{x \to 2}\sqrt[\leftroot{-1}\uproot{3}3]{x^2 + 7x} \ \ \ \ (5)$$ $$= \ \Big( \lim_{x \to 2}x+ \lim_{x \to 2}2 \Big) \cdot \lim_{x \to 2}\sqrt[\leftroot{-1}\uproot{3}3]{x^2 + 7x} \ \ \ \ (4)$$ $$= \ (2+2) \cdot \lim_{x \to 2}\sqrt[\leftroot{-1}\uproot{3}3]{x^2 + 7x} \ \ \ \ (1, \ 2)$$ $$= \ 4 \lim_{x \to 2}\sqrt[\leftroot{-1}\uproot{3}3]{x^2 + 7x}$$ $$= \ 4 \sqrt[\leftroot{1}\uproot{3}3]{\lim_{x \to 2} \Big(x^2 + 7x \Big)} \ \ \ \ (8)$$ $$= \ 4 \sqrt[\leftroot{1}\uproot{3}3]{\lim_{x \to 2} x^2 + \lim_{x \to 2} 7x} \ \ \ \ (4)$$ $$= \ 4 \sqrt[\leftroot{1}\uproot{3}3]{\Big( \lim_{x \to 2} x \Big)^2 + 7 \lim_{x \to 2} x} \ \ \ \ (7, \ 3)$$ $$= \ 4 \sqrt[\leftroot{1}\uproot{3}3]{(2)^2 + 7(2)} \ \ \ \ (2)$$ $$= \ 4 \sqrt[\leftroot{-1}\uproot{1}3]{18}$$

Example 3

$$\lim_{x \to 4} \frac{x}{28}$$ $$= \ \frac{\lim\limits_{x \to 4} x}{\lim\limits_{x \to 4} 28} \ \ \ \ (6)$$ $$= \ \frac{4}{28} \ \ \ \ (1, \ 2)$$ $$= \ \frac{1}{7}$$

Conclusion

As you can see, each of these properties can be applied to fairly complex limits to break them down into smaller, simpler pieces. Each will usually end in applying one of the first two properties listed above to convert a limit into some number. And in the end, you will end up converting all of the limits into numbers. At that point, you will be able to manipulate everything with simple algebra to simplify your answer.

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Solution – Find the values of a and b that make the function differentiable everywhere.

Find all values of a and b that make the following function differentiable for all values of x.

$$f(x) = \begin{cases} bx^2-3 & \mbox{if } x\leq -1 \\ ax+b & \mbox{if } x>-1 \end{cases}$$

When trying to solve a problem like this, there are actually two things you will need to consider for our function f(x).  Obviously, we need to make sure that it’s differentiable everywhere, but this actually implies something else that we will want to consider as well.

Since a function being differentiable implies that it is also continuous, we also want to show that it is continuous.  The reason for this is that any function that is not continuous everywhere cannot be differentiable everywhere.  Once we make sure it’s continuous, then we can worry about whether it’s also differentiable.

Making sure f(x) is continuous everywhere

I’m not going to go into quite as much detail to show the part about making sure the function is continuous because I have already done this, which you can see by clicking here.

To make sure f(x) is continuous at x=-1 we need to make sure that $$\lim_{x \to -1} f(x) = f(-1).$$  Since we have a piecewise function, we will need to consider each one-sided limit, but in this case only the right sided limit will tell us something useful.

$$\lim_{x \to -1^{+}} f(x) = f(-1)$$

$$\lim_{x \to -1^{+}} ax+b = b(-1)^2-3$$

$$a(-1)+b=b-3$$

$$-a+b=b-3$$

$$-a=-3$$

$$a=3$$

So now we know that f(x) will be continuous everywhere as long as a=3.  However, this doesn’t really tell us that f(x) is differentiable everywhere as well.

Making sure f(x) is differentiable everywhere

We now know that we will need to let a=3 in order for this function to be continuous and to have a chance of being differentiable.  As a result, we can say that we are now trying to make this function differentiable everywhere:

$$f(x) = \begin{cases} bx^2-3 & \mbox{if } x\leq -1 \\ 3x+b & \mbox{if } x>-1 \end{cases}$$

We can see that the only place this function would possibly not be differentiable would be at x=-1.  The reason for this is that each function that makes up this piecewise function is a polynomial and is therefore continuous and differentiable on its entire domain.  The only place we may have a problem is when we have to switch between the two functions.

What does it mean for a function to be differentiable?

It means that its derivative exists for all values of x.  In other words, we need to be able to find its derivative no matter what x is.

However, as I mentioned above, in this case we really only need to make sure that we can find the derivative of f(x) when x=-1 since we know it would exist for all other values of x.  By using the definition of a derivative, we need to make sure the following limit exists at x=-1.

$$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

Since we need to check this when x=-1, we can plug in -1 for x.  Therefore, we need to make sure this limit exists:

$$\lim_{h \to 0} \frac{f(-1+h)-f(-1)}{h}$$

I went over this limit definition in greater detail previously.  If you want a refresher on where this is coming from you can find that by clicking here.

Just like when we had to find the limit to make sure that f(x) was continuous, we will need to consider each one sided limit separately in order to find this limit.  And also like when we checked for continuity, each one sided limit is going to require the use of a different section of our piecewise function.

Setting up the limits

When h is slightly less than 0, and we are considering the left sided limit, f(-1+h) would need to be found using the y=bx^2-3 because this would involve inputting x values which are less than -1.

By the same reasoning, when h is slightly greater than 0, and we are considering the right sided limit, f(-1+h) would need to be found using the y=3x+b because this would involve inputting x values which are greater than -1.  Therefore, we need to consider the following one sided limits:

$$\lim_{h \to 0^{-}} \frac{\Big[b(-1+h)^{2}-3\Big]-\Big[b(-1)^{2}-3\Big]}{h}$$

$$\lim_{h \to 0^{+}} \frac{\Big[3(-1+h)+b\Big]-\Big[b(-1)^{2}-3\Big]}{h}$$

Now what do we do with these limits?

Now remember, as I discussed in the lesson about one-sided limits, in order for a limit to exist we need both of its one-sided limits to exist and they need to be equal.  Therefore, in order to show that the derivative of f(x) exists at x=-1, these two one-sided limits need to be equal to each other.  Before setting them equal to each other, first we’ll simplify them a bit.  First the left side limit.

$$\lim_{h \to 0^{-}} \frac{\Big[b(-1+h)^{2}-3\Big]-\Big[b(-1)^{2}-3\Big]}{h}$$

$$=\lim_{h \to 0^{-}} \frac{\Big[b(-1+h)(-1+h)-3\Big]-\Big[b(1)-3\Big]}{h}$$

$$=\lim_{h \to 0^{-}} \frac{\Big[b(1-2h+h^2)-3\Big]-\Big[b-3\Big]}{h}$$

$$=\lim_{h \to 0^{-}} \frac{\Big[b-2bh+bh^2-3\Big]-\Big[b-3\Big]}{h}$$

$$=\lim_{h \to 0^{-}} \frac{b-2bh+bh^2-3-b+3}{h}$$

$$=\lim_{h \to 0^{-}} \frac{bh^2-2bh}{h}$$

$$=\lim_{h \to 0^{-}} \frac{h(bh-2b)}{h}$$

$$=\lim_{h \to 0^{-}} bh-2b$$

$$=-2b$$

And now the right sided limit.

$$=\lim_{h \to 0^{+}} \frac{\Big[3(-1+h)+b\Big]-\Big[b(-1)^{2}-3\Big]}{h}$$

$$=\lim_{h \to 0^{+}} \frac{\Big[-3+3h+b\Big]-\Big[b(1)-3\Big]}{h}$$

$$=\lim_{h \to 0^{+}} \frac{-3+3h+b-b+3}{h}$$

$$=\lim_{h \to 0^{+}} \frac{3h}{h}$$

$$=\lim_{h \to 0^{+}} 3$$

$$=3$$

Now if we set these two simplified versions of the one-sided limits equal to each other, we get

$$-2b=3$$

$$b=-\frac{3}{2}$$

What does this tell us?

So now if we put both pieces together, we know that a=3 will ensure that f(x) is continuous and then making b=-\frac{3}{2} will also make sure f(x) is differentiable at x=-1.  This would in turn make f(x) differentiable for all values of x, or make it differentiable everywhere.

As always, I want to hear your questions!  Go check out my other lessons about derivatives and if you can’t get your question answered, I’d love to hear from you.  Leave a comment below or email me at jakesmathlessons@gmail.com.  If you have questions on this problem and solution or if you have another question you would like to see me answer, just ask it.  Or if you have an entire topic you would like to see me write a lesson about, just let me know.

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Finding Derivatives with Limits

At this point we should have at least a basic understanding of limits and how to find some limits.  However, I have only really discussed limits by themselves and not how they relate to the rest of calculus.  They are very important in calculus because they are used to define the most important calculus topics.

For example, the main topic which will be discussed for quite some time is derivatives.  Derivatives will come up in a lot of different settings, like finding rate of change, instantaneous rate of change, velocity, slope, and a few others.  The main thing to realize is that a derivative is generally used to find out how quickly, or slowly, something is changing.

I will go further into all of these things later, but for now I want to focus on the definition of derivatives and how to find a derivative using the definition.

The definition of a derivative

If we have some function, f(x), we would write “the derivative of f” as f'(x).  And we would define the derivative of f by using this limit:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$

This limit can be a bit confusing, so there’s something I would like to point out before we actually begin working with this limit.  The confusing thing here is that we have x and h in this limit and it looks as if they are both variables.  However, when we find this limit, we can only treat h as a variable.  We will need to treat x as a constant and h as the only variable.

The reason for this is that we are finding the limit as h goes to 0.  This tells us that h is moving in toward 0.  It does not tell us that x is changing at all.  Therefore, when we are working with the limit, we will act as if x is a number, or a constant.  This means that once we find the limit, our answer may have x in it still and this is completely fine since x isn’t the variable in this case.  Now let’s try an example.

Example 1

Consider the function f(x) = 4x^2 - 7x + 12.  We will use the limit definition to find the derivative of this function, but first let’s break it down and consider each part on its own.

Finding f(x+h)

The fist thing we need to find is f(x+h).  This notation basically just means that we need to look at our function f, and plug in (x+h) wherever we see the input.  In other words, we need to replace all of the x‘s in the function with (x+h)‘s.  So,

$$f(x+h) = 4(x+h)^2 – 7(x+h) + 12.$$

Then we will want to expand this out so it’s easier to work with.  Remember (x+h)^2 is the same as (x+h)(x+h), which means we need to foil it.

$$f(x+h) = 4(x+h)(x+h) – 7(x+h) + 12$$

$$=4(x^2 + xh + xh + h^2) – 7(x+h) + 12$$

$$=4(x^2 + 2xh+ h^2) – 7(x+h) + 12$$

$$=4x^2 + 8xh+ 4h^2 – 7x – 7h + 12$$

Since there aren’t any like terms we will leave it at that for now.

Putting it all together

Now we can put that into the rest of the equation.  Since we now know f(x+h) and f(x), we can plug those into the equation.  I would recommend surrounding each of them with a set of parenthesis so you don’t forget to distribute the negative sign in front of the f(x).  This is a very common mistake, so be careful not to forget that because it will give you the wrong answer.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$

$$= \lim_{h \to 0} \frac{(4x^2 + 8xh+ 4h^2 – 7x – 7h + 12) – (4x^2 – 7x + 12)}{h}$$

Solving the limit

When I first see a limit, the first thing I usually consider is whether we can simply plug in 0 for h. Essentially, I try to treat this function as if it were continuous at h=0 (remember h is the variable here).

However, if we do this here we will get 0 on the denominator.  Since you cannot divide by 0, this will not work.  So our strategy will be to simplify this fraction to a point where we can plug in 0 for h.  The simplest way to do this is to rearrange the numerator so that we can cancel an h from the numerator and denominator and get rid of our fraction all together.

$$f'(x)= \lim_{h \to 0} \frac{4x^2 + 8xh+ 4h^2 – 7x – 7h + 12 – 4x^2 + 7x – 12}{h}$$

$$= \lim_{h \to 0} \frac{8xh+ 4h^2 – 7h}{h}$$

At this point I would like to point something out. Notice, after simplifying the numerator of the fraction, each term remaining contains an h in it.  This is important because it allows us to factor the h out and cancel it with the h in the denominator, getting rid of the fraction.  This will be an extremely common strategy to use for finding the derivative of a function using the limit definition.

$$f'(x)= \lim_{h \to 0} \frac{h(8x+ 4h – 7)}{h}$$

$$= \lim_{h \to 0} 8x+ 4h – 7$$

Now we have simplified to a point that we can solve this limit by plugging 0 in for h.

$$f'(x)= 8x+ 4(0) – 7$$

$$= 8x – 7$$

So we have just shown that if f(x)=4x^2-7x+12, then f'(x)=8x-7.  We will later learn shortcuts like the product rule, quotient rule, and chain rule that will make finding derivatives like this much simpler and faster, but you will need to know how to find these using the limit definition.  The pattern shown in this problem is a common one.  It won’t work for all derivatives, but it’s a good thing to try first.

  • It’s generally a good idea to see if you can reengage the top of the fraction in such a way that every term has h as a factor.
  • Then you can factor out the h, and cancel it with the h on the bottom of the fraction.
  • This usually leaves you with a function that you can directly plug 0 into h and simplify from there, leaving you with a function that doesn’t contain any h‘s, but usually contains x‘s.

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I would recommend checking out the other material I have about derivatives.  As I mentioned before, there are several shortcuts and methods that make this whole process a lot easier.  Go check out what I’ve written about on the derivatives page.  If you have a question that isn’t answered there just let me know by emailing me at jakesmathlessons@gmail.com.  I’ll do my best to answer any questions you send me and I may even post a lesson or full solution on it!

Continuity

Continuity is a relatively simple concept, but problems that require proving it can be a little tricky. Essentially, a continuous function is one that you can draw all in one motion without picking up your pencil. This is one explanation of what it means for a function to be continuous that I like because it doesn’t take any mathematical definitions or proofs to understand. Any holes or gaps in a function’s graph would be a discontinuity and would mean that the function is not continuous.

The limit definition of continuity

By definition, a function f(x) is continuous at x=a if $$\lim_{x \to a} f(x) = f(a).$$

Let’s think about what this equation is saying. The left side of this equation is something that we’ve already dealt with, limits. It’s asking us to find out what y value we close in on as we travel along our function and close in on x=a. Keep in mind, x is a variable here which represents the input of our function, f(x), and a is a constant. This means that a represents some specific number, which could be any number.

The right side of the equation is simply asking us to plug that same a value into f(x) and take the y value we get out.

In total, the above equation says that if we travel along a function and close in on a specific x value, we should close in on the same y value we would get if we simply plugged that x value into the equation.

In other words, as we travel along a function toward a specific x value, our y value will also go toward the y value of the function at that point. If that is the case, then the function is continuous at that specific point. If we can say that a function is continuous at every single possible value we could put in for a, then we can say that the function is continuous for all x. If this is true then we can draw our entire function in one motion without picking up our pencil!

Let’s do a couple examples.

Example 1

Remember back in the first lesson about limits, Limits – Intro, I said I would go back to discussing the importance of our limit in the first example giving us the same value as when we plugged x=2 into the equation?  I would like to go into that further.

continuity
Figure 1.1

The function we were considering was f(x) = x^2 and we were finding

$$\lim_{x \to 2} x^2.$$

After looking at the graph of this function, shown in Figure 1.1, we saw that

$$\lim_{x \to 2} x^2 = 4.$$

I also pointed out that plugging in x=2 directly into the function also returns a y value of 4. In other words, we know $$f(2) = 4.$$ Therefore, we know

$$\lim_{x \to 2} x^2 = 4 = f(2)$$

$$\lim_{x \to 2} x^2 = f(2).$$

Notice this is exactly like the definition of what it means for a function to be continuous at a point. If you replace a with 2, replace f(x) with x ^2, and replace f(a) with 4, we have just shown that y=x^2 is a function that is continuous at x=2.

Example 2

The next example I want to discuss also goes back to a function we have already looked at. Referring back to Limits – Intro we will consider f(x) shown in Figure 1.2.

Figure 1.2

The question we will answer here is whether this function is continuous at x=1 or not. What do you think?

There are a few different ways we can answer this question. The simplest would be to simply look at the graph of the function and think about whether we can draw that function at and around x=1 without picking up our pencil. As you can see, there is clearly a hole at x=1 where we would need to pick up our pencil, and add a single point at (1, \ 4).

As a result, it is probably safe to say that this function is not continuous at x=1. However, we want to be able to show this using the actual definition of what it means for a function to be continuous.

Remember, for this function, which we are calling f(x) in this case, we need to be able to show that $$\lim_{x \to 1} f(x) = f(1).$$

If we can show this equation to be true, then f(x) is continuous at x=1, and if it’s not true then the function is not continuous at x=1. Luckily, back when I first used this function as an example in the Limits – Intro lesson, we found that $$\lim_{x \to 1} f(x) = 2.$$ Therefore, we just need to find out if f(1) is also 2 and we can prove that this function is continuous or not continuous.

Looking at the graph again, we see that this function has a hole at x=1 and includes the point (1, \ 4). In other words, if we plug 1 into f(x) as our x value, we get a y value of 4 out. This is the same as saying f(1) = 4.

Now, at this point we have figured out $$\lim_{x \to 1} f(x) = 2 \ and$$ $$f(1) = 4.$$

Therefore, $$\lim_{x \to 1} f(x) \neq f(1)$$ and we can say that f(x) is not continuous at x=1.

More Examples

Find the values of a and b that make f continuous everywhere.

$$f(x) = \begin{cases} \frac{x^2-4}{x-2} & \mbox{if } x<2 \\ ax^2-bx+3 & \mbox{if } 2\leq x<3 \\ 2x-a+b & \mbox{if } x\geq 3 \end{cases}$$

To see the solution to this problem, click here.

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There are also several other lessons and problems on my limits page.  It would be a good idea to get some practice with limits.  A lot of other more complex topics in calculus are based around limits so they are important to understand.  If you can’t find the topic you want to read about just let me know by emailing me at jakesmathlessons@gmail.com and I’ll do my best to answer your question!

Squeeze Theorem

The Squeeze Theorem is a useful tool for finding complex limits by comparing the limit to two much simpler limits. Squeeze Theorem tells us that if we know these three things:

$$1. \ \ \ g(x) \leq f(x) \leq h(x)$$

$$2. \ \ \ \lim_{x \to a} g(x) = L$$

$$3. \ \ \ \lim_{x \to a} h(x) = L$$

Then we also know that

$$\lim_{x \to a} f(x) = L$$

Keep in mind, requirement number 1 above only needs to be true around x=a. It does not need to be true at x=a and it also does not need to be true when we get far away from x=a. This is because limits only care what is happening to a function when we get infinitely close to a given x value, not at that x value and certainly not far away from that x value.

Example 1

You will typically see this used when you are finding the limit of a function which is made up of two functions being multiplied together and one of them is a trig function. For example, consider

$$\lim_{x \to 0}x ^4 sin \bigg( \frac{1}{\sqrt{x}} \bigg).$$

At first glance, this limit looks terribly complicated. Especially because the function f(x) = x ^4 sin \Big( \frac{1}{\sqrt{x}} \Big) doesn’t even exist at x=0 (plugging in x=0 directly causes you to divide by 0, which can’t be done). However, if we use Squeeze Theorem, we can find this limit by finding two other limits that are much easier to find.

Squeeze Theorem says that we will need to find one function that is greater than or equal to f(x) around x=0 (because we are finding the limit as x \to 0) and another function that is less than or equal to f(x) around x=0. It is not always obvious which functions to use for comparison, but any function that is made up of sine or cosine of something multiplied by something else will usually follow the same pattern, so this is a good thing to try first.

What’s the pattern?

Consider the function y = sin(x). No matter what you plug in for x you will always get a y value between -1 and 1. This will be true regardless of what you are taking the sine of. Even if you were to take sin\Big( \frac{1}{\sqrt{x}} \Big). No matter what you put in for x this will give you something between -1 and 1, except if you try to put in x=0 (remember you can’t divide by 0).

But remember, with limits it doesn’t matter what happens at the x value we are approaching, only what is happening around that point. Therefore, it doesn’t matter that this function isn’t defined at x=0 because it is defined for all x values near x=0.

The point I’m trying to make is that sine of anything can be bound between -1 and 1, so we can say:

$$-1 \leq sin(x) \leq 1, \ and$$

$$-1 \leq sin \bigg( \frac{1}{\sqrt{x}} \bigg) \leq 1$$

How does this relate to our example?

Now, the trick that makes this whole thing work is the fact that multiplying both sides of an inequality by a positive number maintains the inequality. Therefore, we can also multiply both sides of the inequality by a function that always outputs a positive number (or 0).

For example, we can multiply both sides by x^4, or all three “sides” in this case. We can do this because any number raised to an even power will always be positive (or 0). This gives us

$$x^4 \cdot \Bigg[ -1 \leq sin \bigg( \frac{1}{\sqrt{x}} \bigg) \leq 1 \Bigg]$$

$$x^4 \cdot (-1) \ \leq \ x^4 \cdot \Bigg(sin \bigg( \frac{1}{\sqrt{x}} \bigg) \Bigg) \ \leq \ x^4 \cdot (1)$$

$$-x^4 \ \leq \ x^4 sin \bigg( \frac{1}{\sqrt{x}} \bigg) \ \leq \ x^4$$

Now, by using the Squeeze Theorem, since x^4 sin \bigg( \frac{1}{\sqrt{x}} \bigg) is trapped between -x^4 and x^4 near 0, if we find that if

$$\lim_{x \to 0} -x^4 = \lim_{x \to 0} x^4$$

then that will also be the answer for the limit we are looking for. Consider the graph of y=-x^4 and y=x^4 around x=0 shown below which was graphed using Desmos.

squeeze theorem

You can see that for both of these functions, as we travel along each function from both sides and get closer and closer to x=0, we get closer and closer to a y value of 0. This tells us that

$$\lim_{x \to 0} -x ^4 = 0, \ and$$

$$\lim_{x \to 0} x ^4 = 0$$

Putting it all together

These two facts in combination with the inequality we showed earlier: $$-x^4 \ \leq \ x^4 sin \bigg( \frac{1}{\sqrt{x}} \bigg) \ \leq \ x^4,$$ tells us that

$$\lim_{x \to 0}x ^4 sin \bigg( \frac{1}{\sqrt{x}} \bigg) = 0.$$

Note: Compare these last 4 statements to the 4 statements at the beginning of this lesson.  The inequality is exactly the first thing I said we were looking to be able to show and the two limits are exactly like part 2 and 3.  So it’s no surprise that these three pieces combine to prove the limit we were trying to find.

Example 2

\mathbf{\lim\limits_{x \to \infty} \frac{sin(x)}{x}} | Solution

The Squeeze Theorem is an important application of limits in calculus, but I have written several other lessons and problem solutions about limits!  You should go check out my limits page to see what else I’ve done.  If I haven’t answered any questions you have on that page, let me know by sending me an email at jakesmathlessons@gmail.com and I’ll do my best to address your questions.

Also, you can use the form below to be added to my email list and I will send you my calculus 1 study guide as a FREE welcome gift!

One-sided Limits

To end the last lesson, Limits – Intro, I mentioned that a limit would not exist if you did not approach the same y value as you approach a given x value from both the left and right.  For example, consider f(x) shown in Figure 1.2 again (shown again below), but this time let’s find:

$$\lim_{x \to -1}f(x).$$

In the previous section I mentioned approaching the given x value from the left and the right side to find the limit. This is necessary when finding two-sided limits (usually just referred to as “limits”), but let’s consider each side separately. In other words, we will consider what y we get close to as we approach x=-1 from the left side as one problem and from the right side as a separate problem.

Approaching from the left side

First we will approach x=-1 from the left side. This is denoted like this:

$$\lim_{x \to -1 ^{-}}f(x).$$

Notice the little ^- to the right of the -1. This tells us the we are approaching x=-1 only from the negative side, or the left side. As shown below in Figure 1.3, as we approach x=-1 from the left side, we get closer to y=2.

Figure 1.3

Therefore, we can say that

$$\lim_{x \to -1 ^{-}}f(x) = 2.$$

Approaching from the right side

Similarly, if we only consider what y value we approach as we get close to x=-1 from the right side, or the positive side, we are finding:

$$\lim_{x \to -1 ^{+}}f(x).$$

Just like before, we only want to consider approaching this specific x value from one side:

right sided limit
Figure 1.4

As you can see, if we start on this function to the right of x=-1 and we move toward x=-1 along the function, we get closer and closer to a y value of 4. Therefore, we can say that this one-sided limit has a value of 4, or

$$\lim_{x \to -1 ^{+}}f(x) = 4.$$

Putting them together

Now, we have found both one sided limits of this function around x=-1. As we approach x=-1 from the left side, we get closer to y=2, but as we approach x=-1 from the right side, we get closer to y=4. Since we get infinitely close to two different y values depending on whether we approach x=-1 from the left side versus the right side, this two-sided limit actually does not exist. So,

$$\lim_{x \to -1}f(x) \ \ DNE.$$

Why do we need to consider each one sided limit separately?

This is an important thing to remember. In order to find any two-sided limit, you will instead find each one sided limit. If both one-sided limits are the same, then the two-sided limit will also be that same value. However, if the one-sided limits are different, the two-sided limit does not exist. In other words, $$if \ \lim_{x \to a ^{-}}f(x) = \lim_{x \to a ^{+}}f(x) = b, \ then \ \lim_{x \to a}f(x) = b.$$ $$And \ if \ \lim_{x \to a ^{-}}f(x) \neq \lim_{x \to a ^{+}}f(x), \ then \ \lim_{x \to a}f(x) \ does \ not \ exist.$$

Another method that can be used to evaluate one-sided limits if you don’t have a graph of the function available is using the properties of limits. You can learn more about the limit properties here.

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For more on limits go check out my limits page.  There’s a list of lessons and practice problems there all about limits.  Take a look to get some more practice with limits.  If you have a specific topic or problem you’re looking for and can’t find it, then email me at jakesmathlessons@gmail.com.  Send me your questions and I’ll be sure to point you in the right direction to get them answered.  I may even write a lesson and post it to make sure your question gets answered!