Does anyone know l'Hôpital's rule for limits? I was doing an assignment and the middle 2 questions are l'Hôpital's rule questions and we haven't even done this in class yet. These are 1 or 2 chapters away yet we are expected to do them and I have no idea how to do these because we haven't learned yet so I was hoping someone here can help me.
First question:

An incorrect use of l'Hôpital's rule is illustrated in the following limit computation. Explain what is wrong and find the correct value of the limit.
$$\lim_{x\to\pi/2}   \frac{\sin(x)}{x}    = \lim_{x\to\pi/2}  \frac{\cos(x)}{1}=0$$

Second question is:

Find $\lim\limits_{x\to-\infty} xe^x$ using l'Hôpital's rule. Using this information does the function $xe^x$ have a horizontal or vertical asymptote? state the equation of the asymptote.

I have done about 15 pages worth of questions and some I knew because I studied extra things on spare time but these are just hard.
 A: I would advise looking ahead in your book and just check out what l'Hôpital's rule says. This site is useful for answering a specific question, and helpful to use as a supplement to learning in class or from a book, but not really designed with teaching of a broader subject. I would also check with your instructor, as it is possible that those two questions were mistakenly included in this assignment.
That said, from Wikipedia's entry:
In its simplest form, l'Hôpital's rule states that for functions $f$ and $g$:
$$\text{If }\lim_{x\to c} f(x)=\lim_{x\to c} g(x)=0 \text{ or } \pm \infty\text{ and }\lim_{x\to c} \frac {f'(x)}{g'(x)} \text{ exists,}$$
$$\text{then } \lim_{x\to c} \frac {f(x)}{g(x)}=\lim_{x\to c} \frac{f'(x)}{g'(x)}\ .$$ 
From this, you should be able to see why the application of l'Hôpital's is incorrect to apply to $\lim_{x\to \pi/2}\sin(x)/x$. What is the requirement of the rule? Does your limit satisfy that requirement?
For the second question, you can proceed as follows:
$$\lim_{x\to -\infty} x e^x = \lim_{x\to \infty} -x e^{-x} = \lim_{x\to\infty} \frac {-x}{e^x}\ ,$$
which then succumbs to l'Hôpital's rule.
Again, I would encourage you to check with your professor on these questions. If you haven't covered l'Hôpital's rule yet, my guess is that they did not mean to include them in this assignment.
A: l'Hospital's rule only applies for
\begin{equation}
 \lim \frac{f(x)}{g(x)}
\end{equation}
if $\lim f(x)$ and $\lim g(x)$ is $\pm\infty$, or if $\lim f(x) = \lim g(x) = 0$.
And if they don't, then the limit is trivially found to start with, as you can see in $$\lim_{x\to\pi/2} \frac{\sin(x)}{x} = \frac{\sin(\pi/2)}{(\pi/2)} = \frac{2}{\pi}$$ Neither nominator or denominator goes to zero or infinity.
For the second question, are you sure you got it right? l'Hospital doesn't apply as it reads right now.
Edit: The sign was lost in the edit when i answered this. For the current expression, l'Hospital holds, and its straightforward to use it.
A: $$
\lim_{x\to-\infty} xe^x = \lim_{x\to-\infty} \frac{x}{e^{-x}}.
$$
The numerator approaches $-\infty$ and the denominator approaches $\infty$, so L'Hopital's rule is applicable.
(The other parts of your question seem to have been dealt with already.)
