Question

We have `f:RR->RR,f(x)=(x+2)e^(-|x|)`How to solve this limit? `lim_(x->0)(f(x)-f(0))/(x);x>0`

Answer 1

The limit does not exist.

Explanation:

The left and right limits at `0` are different.

The quickest way to see this is to note that this limit is the definition of `f'(0)`.

But for this function we have

`f(x) = {((x+2)e^x,"if",x >= 0),((x+2)e^(-x),"if",x < 0):}`

The right derivative is `e^x+(x+2)e^x` which is `2` at `x=0`

while the left derivative is `e^(-x)-(x+2)e^-x`, which is `-2` and `x=0`.

Using technology

The graph of `f` is shown below.

graph{y=(x+2)e^(-abs(x)) [-5.213, 5.884, -2.11, 3.44]}