TITLE: Piecewise function analysis
QUESTION [1 upvotes]: $f(x)=\left\{
\begin{array}{ll}
      \frac{e^{-x}-1}{x}, & x>0 \\
      \frac{x}{2}+1, & x\leq 0 \\
\end{array} 
\right. $
a) At which intervals are f strictly increasing and at what intervals are f strictly decreasing.
Should I just find the derivative of both of the functions? If so, I get that the function is increasing in the intervals $(-\infty, 0)$ and $(0, \infty)$. Is this right, or can I just right $(-\infty, \infty)$?
b) Determine any local and global extreme values for f.
When graphing the function I don't see any local or global extreme values. The whole function is not bounded, so I can't look at the values in the endpoints. But my teacher says there are extreme values. So I'm confused
c) Determines if f has asymptotes.
I know there is at least one asymptote, y=0, given the first function. But I was told there is one oblique asymptote also. But how?

REPLY [1 votes]: a) you have seen that $f$ is increasing at $(-\infty, 0)$ and $(0, \infty)$. Now you need to see whether $f(0) > f(a)$ for $a < 0$ to conclude $f$ is strictly increasing everywhere. Hint: consider $\lim_{x \to 0+}f(x)$.
b) A local extreme value needs to satisfy $f'(a) = 0$ only if $f$ is differentiable there. $f$ is not continuous at $0$, let alone differentiable. Have a better look at $f$'s behaviour around 0.
c) What does $\frac{x}{2} + 1$ look like for $x$ very small?
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