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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Practice on L'Hopital's Rule}
\centerline{April 5, 2018}
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\noindent
{\it Background}
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L'Hopital's Rule is a method for determining whether indeterminate form
limits such as $0/0$ limits or $\infty/\infty$ limits exist and computing them
if they do.  It can 
also be applied in other indeterminate cases such as $\infty\cdot 0$ and $1^\infty$
by using additional algebraic maneuvers and/or properties of logarithms. The basic
statement is this:
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\noindent
[L'Hopital's Rule]  \emph{If $\frac{f(x)}{g(x)}$ is $0/0$ or $\infty/\infty$ as $x \to c$ 
($c$ can be finite or infinite, the limits can be one- or two-sided) and 
$$\lim_{x \to c} \frac{f'(x)}{g'(x)} = L$$
exists, then 
$$\lim_{x \to c} \frac{f(x)}{g(x)} = L$$
as well.}  
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Note:  We are \emph{not} applying the Quotient Rule to compute the derivative of 
$\frac{f(x)}{g(x)}$ here; L'Hopital's Rule uses the limit of the quotient of 
the derivatives of the numerator and denominator separately:  $\frac{f'(x)}{g'(x)}$.  
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\noindent
{\it Questions}
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Use L'Hopital's Rule as indicated to compute the following limits
\begin{enumerate}
\item[(1)] $\lim_{x\to 1} \frac{x^3 - 1}{\sqrt{x} - 1}$
\item[(2)] $\lim_{x \to \infty} \frac{\ln(x)}{x}$
\item[(3)] For $\infty \cdot 0$ forms $\lim_{x \to c} f(x)\cdot g(x)$, note that
rewriting
$$f(x) \cdot g(x) = \frac{f(x)}{1/g(x)}$$
converts to a $\infty/\infty$ form (if $\lim_{x \to c} g(x) = 0$ and $\lim_{x \to c} f(x) = \infty$). 
Hence you can apply L'Hopital to the rearranged form. 
Apply this idea to compute
$$\lim_{x\to 0^+} x\ln(x)$$ 
\item[(4)] If $\lim_{x\to c} f(x) = 1$ and $\lim_{x\to c} g(x) = \infty$, then 
$\lim_{x \to c} f(x)^{g(x)}$ is also an indeterminate form called a $1^\infty$ form.  
The standard approach for these is to take natural logs and use the fact that 
$$\ln\left(f(x)^{g(x)}\right) = g(x) \ln(f(x))$$
The limit $\lim_{x \to c} g(x) \ln(f(x))$ is then an $\infty \cdot 0$ form and we 
can proceed as in (3).  If 
$$\lim_{x \to c} g(x) \ln(f(x)) = L$$
then the continuity of $\ln$ implies $\lim_{x\to c} f(x)^{g(x)} = e^L$.
Apply this approach to compute 
$$\lim_{x \to 0} (1 + x)^{1/x}$$
\end{enumerate}
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\caption{The region for problem 4.}
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