Tuesday, July 14, 2015

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

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