Math Help - Limits problem

1. Limits problem

Hi everyone,

I've been struggling with 2 limits problems that couldn't be solved by me, and I'm asking for help.

please help with the attached problems, basically I need to find a example for a function that does NOT follow the rules specified in the question.

2. For the first one:

take $f(x)= 1, g(x)= e^{-x}$

$\lim_{x\to\infty} |\frac{1}{e^{-x}}|=\infty$ but $\lim_{x\to\infty}|1-e^{-x}|= 1$.

3. The second one is kind of weird, a limit $\lim_{x\to a}f(x)$ only exists when $\lim_{x\to a^-}f(x)$ and $\lim_{x\to a^+}f(x)$ both exist and are equal.

So if we take $f(x)= 4x$ then $\lim_{x\to 1}f(x)= 4$ and $\lim_{x\to 0}4\cdot \frac{|x|}{x}$ is undefined since $\lim_{x\to 0^+}4\cdot \frac{|x|}{x}=4$ and $\lim_{x\to 0^-}4\cdot \frac{|x|}{x}= -4$

Is this the kind of counter-example you're looking for ?