Given function: $$f(x)=|x+2|e^{-\frac{1}{|x|}}$$
- Find constants $a, b$ and $c$ such that $f(x)= ax +b+\frac{c}{x}+ \sigma(\frac{1}{x})$ when $x \rightarrow \infty$ and $x \rightarrow -\infty$?
- Examine the function and then graph it.
- Examine function differentiability.
- Determine the equation of the tangent on the function graph at the point $(1, f(1))$
I understand how parts 2. and 3. are done but have never encountered 1. and 4.
2.
$$f(x)=\begin{cases}-(x+2)\cdot e^{\frac{1}{x}}, & x<-2\\(x+2)\cdot e^{\frac{1}{x}}, & -2\le x<0\\(x+2)\cdot e^{-\frac{1}{x}}, & x>0\end{cases}$$
Domain for these functions is $x \in \mathbb{R} \setminus {0}$ and I would graph only parts of the functions corresponding to their domain.
- I check the critical points $x=-2$ and $x=0$ but since $x=0$ isn't part of the domain (and therefore $f$ is not continuous at that point) I would exclude this part. For $x=-2$, I swap it into the left derivative (of the first function) and do the same for the right derivative (of the second function). I get that they're not matching for $f'(-2)$ and conclude that the function is not differentiable at $x=-2$.
These are my brief explanations of thought process for 2. and 3.
The ones remaining are 1. (which I haven't encountered before but looks like a Taylor expansion) and 4.