1. ## limits

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2. Originally Posted by hinchcm
If there is ANYONE out there who is a kind enough person to help me with this (and knows how to do it!) I would be VERY grateful!! I only have one more day to work on these problems!! Thank you so much!

For the following functions:
a) y = xe^(1/x)
b) y = xe^(-x)

I need to find:
1) vertical and horizontal asymptotes.

Mr F says: See main reply.

2) relative extrema
3) inflection points
4) intervals where the function is increasing/decreasing
5) where the function is concave up/down

Mr F says: Do you know how to find dy/dx? Do you know how dy/dx relates to the above?

Be sure to evaluate limits at the borders of the domain of the function (e.g. infinity, negative infinity, one sided limits around vertical asymptotes...)
For $\displaystyle y = x e^{1/x}$ note that:

$\displaystyle \lim_{x \rightarrow 0^+} x e^{1/x} = + \infty$.

$\displaystyle \lim_{x \rightarrow 0^-} x e^{1/x} = 0$.

$\displaystyle \lim_{x \rightarrow +\infty} x e^{1/x} = + \infty$.

$\displaystyle \lim_{x \rightarrow -\infty} x e^{1/x} = -\infty$.

Note: $\displaystyle x e^{1/x} = x \left( 1 + \frac{1}{x} + \frac{1}{2} \frac{1}{x^2} + \, .....\right)$