Extra credit questions on L'Hopital's Rule

This is worth 5 points extra credit if done properly and turned in by May 4.

L'Hopital's Rule is a valuable tool for computing certain limits, limits of a particular indeterminate form. These forms are of type $0/0$ and $\infty/\infty$.

The question is to determine if the following has a limit, and if it does what is it:

$$~ \lim_{x \rightarrow c} \frac{f(x)}{g(x)}. ~$$

The basic idea is if you plug in "c" and you get the indeterminate forms above, then if

$$~ L = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}, ~$$

then this same $L$ is the answer to the first question.

In julia we can approach this numerically. First, we load the MTH229 package for plotting and its easy-to-use derivative operation:

using MTH229

Let's look at a favorite example – whose answer is known by other means – $\sin(x)/x$ at $c=0$.

f(x) = sin(x)
g(x) = x
c = 0
f(c)/g(c)			# NaN is indeterminate
f'(c) / g'(c)		# We can plug in to get the limit!

So the limit is $1$.

Some times, we get the $f'(c)/g'(c)$ is also of indeterminate form. In that case, if $f''(x)/g''(x)$ has a limit at $c$, then so will $f(x)/g(x)$, etc. If higher-order derivatives are need, the syntax is f'', f''', ...

Exercises

• Use L'Hopital's rule to evaluate the limit or state that the limit does not exist or that the rule does not apply.

$$~ \lim_{x \rightarrow 0}\frac{\sin(4x)}{x^2 + 3x + 1}. ~$$

• Use L'Hopital's rule to evaluate the limit or state that the limit does not exist or that the rule does not apply.

$$~ \lim_{x \rightarrow 0}\frac{1 - \cos(x) - \sin^2(x)}{\sin(x)}. ~$$

• Compute the following limit with the aid of L'Hopital's rule.

$$~ \lim_{x \rightarrow 0} \frac{\tan^{-1}(x)}{\sin^{-1}(x)}. ~$$

• Use L'Hopital's rule to evaluate the limit or state that the limit does not exist or that the rule does not apply.

$$~ \lim_{x \rightarrow 0}\frac{(1-\cos(x))^2}{x^4 + x^2}. ~$$

• Rewriting functions. The form $f(c)\cdot g(c) = 0 \cdot \infty$ can be rewritten to become $0/0$ by considering the ratio $f(c)/(1/g(x))$.

Compute the following limit with the aid of L'Hopital's rule.

$$~ \lim_{x \rightarrow 1} \tan(\frac{\pi x}{2}) \ln(x). ~$$