Advanced Math Solutions – Limits Calculator, Squeeze Theorem

What happens when algebraic manipulation does not work to find the limit? Give the squeeze theorem, also known as the sandwich theorem, a try! The squeeze theorem helps you find the limit of a function by comparing the limits of two simpler functions that are the lower and upper bounds.

The Squeeze Theorem:

What does the Squeeze Theorem mean?

Given a function, f(x), take two simpler functions, g(x) and h(x), that are a higher and lower bound of f(x). If the limit of g(x) and h(x) as x approaches c are the same, then the limit of f(x) as x approaches c must be the same as their limit because f(x) is squeezed, or sandwiched, between them.

Here is an image to help better understand the theorem:

Here we will work out the first problem step by step (click here):

1. Try Substitution

When we substitute 0 for x, we get an undefined answer.

2. Find g(x)and h(x)

We know that \sin(x), it doesn’t matter what x is, is between -1 and 1. We multiply the inside, f(x), by x^2, to get our original function. We multiply the outside functions, g(x) and h(x), by x^2 too.

3. Substitution for the outer limits

We substitute in 0 for x in g(x) and h(x) to find their limits. Since their limits as x approaches 0 both equal 0, then by the squeeze theorem, the limit of f(x) as x approaches 0 is also 0.

Here is an image to better understand the solution to the problem:






g(x)=-x^2
f(x)=x^2\sin(\frac{1}{x^2})
h(x)=x^2








Here’s another example (click here):

Last example (click here):

The squeeze theorem is a very useful theorem to quickly find the limit. However, finding the upper and lower bound functions can be hard. Sometimes graphing f(x) in order to see what the function approaches at x can be helpful when deciding what the lower and upper bounded functions should be.

Until Next Time,

Leah

Advanced Math Solutions – Limits Calculator, The Chain Rule

In our previous post, we talked about how to find the limit of a function using L'Hopital's rule. Another useful way to find the limit is the chain rule. When the chain rule comes to mind, we often think of the chain rule we use when deriving a function. However, the chain rule used to find the limit is different than the chain rule we use when deriving.

The Chain Rule:

What does the chain rule mean?

Given a function, f(g(x)), we set the inner function equal to g(x) and find the limit, b, as x approaches a. We then replace g(x) in f(g(x)) with u to get f(u). Using b, we find the limit, L, of f(u) as u approaches b. The limit of f(g(x)) as x approaches a is equal to L.

That sounds like a mouthful. Here we will go step by step for the first problem to better understand the chain rule (click here):

1. Find g(x) and f(u)

Since g(x) is the inner function, we set g(x)=\sin(x^2). We then replace the g(x) in f(g(x)) with u. Thus, f(u)=e^u.

2. Find the limit, b, of g(x)

3. Find the limit, L, of f(u)

We now get our answer:

Here is another example (click here):

Last example (click here):

Understanding the chain rule may be a little difficult, but once you practice some problems, which you can find on our website, the chain rule becomes much easier.

Until next time,

Leah

Advanced Math Solutions – Limits Calculator, L’Hopital’s Rule

In the previous posts, we have talked about different ways to find the limit of a function. We have gone over factoring, functions with square roots, and rational functions. What happens when none of those options work? That is when L’Hopital’s Rule comes in.

L’Hopital’s Rule:
If \lim_{x\to a}(\frac{f(x)}{g(x)})=\frac{0}{0}  or  \lim_{x\to a}(\frac{f(x)}{g(x)})=\frac{\pm\infty}{\pm\infty}, where a is finite or \pm\infty,
Then \lim_{x\to a}(\frac{f(x)}{g(x)})=\lim_{x\to a}(\frac{f^{'}(x)}{g^{'}(x)})

What does L’Hopital’s Rule mean?

If f(x) and g(x) are differentiable and if the limit of \frac{f(x)}{g(x)} as x approaches a is \frac{0}{0} or \frac{\pm\infty}{\pm\infty}, then we take the derivative of the numerator and the derivative of the denominator,  \frac{f^{'}(x)}{g^{'}(x)} . The limit of \frac{f(x)}{g(x)} as x approaches a is equal to the limit of \frac{f^{'}(x)}{g^{'}(x)} as x approaches a.

When can you use L’Hopital’s Rule?

1. When the limit is in the indeterminate form of \frac{0}{0} or \frac{\pm\infty}{\pm\infty}
2. Sometimes when the limit is in the other indeterminate forms
3. When the function has an exponent and the limit is an indeterminate form 

Here we will work out the first problem step by step (click here):

Try Substitution

     We get an indeterminate form of \frac{0}{0}

Take the derivative of the numerator:

Take the derivative of the denominator:

Simplify the function:
\lim _{x\to \:-3}\left(\frac{\frac{x}{\sqrt{x^2+16}}}{1}\right)=\lim _{x\to \:-3}\left(\frac{x}{\sqrt{x^2+16}}\right)

Substitution:
\frac{-3}{\sqrt{\left(-3\right)^2+16}}=-\frac{3}{5}


Here’s another example (click here):

Last example (click here):

L’Hopital’s rule is an easy way to find the limit, as long as the derivatives aren’t too tedious. It is important to remember to double check your work. I found myself a lot of times making simple errors, like forgetting the chain rule or changing signs. Next week we will talk about the chain rule for limits.

Fun fact: It is believed that Johann (John) Bernoulli discovered L’Hopital’s Rule.

Until next time,
Leah

Practice, practice, practice

Math can be an intimidating subject. Each new topic we learn has symbols and problems we have never seen. The unknowing is what psyches ourselves out. It is important to remember that we all have the ability to learn. Math doesn’t have to be scary! Once you learn the topic, it is important to go a few problems step by step to understand what is going on. Afterwards, you must practice! What they say is true, “practice makes perfect.” The more you practice the better you get.

Try Symbolab’s practice!

What is practice?

Practice is an interactive component of Symbolab. In practice, subjects are broken down into topics. The topics are categorized under difficulty. Select the topic, which you need practice in, work out the first problem, and enter in the solution. If you need help solving the problem, click “Hint”. Each hint will show you a step to solving the problem. Use a hint when your answer is wrong or try the problem again. It is recommended that you get five problems solved correctly, before continuing on to the next topic.

After you finished all the topics in the section, take a quiz to test your skills! Answer 10 questions without hints and see if you are ready for a more advanced topic. At the end of the quiz you will get your results.

Why should I use practice?

Practice is a great resource that helps you better understand math topics.  You are given problems that will cover all different types of problems under the topic. If you get stuck along the way, you have hints that will take you through the problem step by step. There is no better way to excel at math than to study hard and practice, practice, practice. It’s a great way to review before a test. Practice won’t be free for long, so make sure you register for it now!

Everyone has to ability to learn math. Practice is a free way to take advantage of your ability and learn! Math doesn’t have to be scary. Practice, practice, practice and the easier it will become.

Until next time,

Leah