
2 Attachment(s)
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.
Please advise if you can, thanks in advance.

For the first one:
take $\displaystyle f(x)= 1, g(x)= e^{x}$
$\displaystyle \lim_{x\to\infty} \frac{1}{e^{x}}=\infty$ but $\displaystyle \lim_{x\to\infty}1e^{x}= 1$.

The second one is kind of weird, a limit $\displaystyle \lim_{x\to a}f(x)$ only exists when $\displaystyle \lim_{x\to a^}f(x)$ and $\displaystyle \lim_{x\to a^+}f(x)$ both exist and are equal.
So if we take $\displaystyle f(x)= 4x$ then $\displaystyle \lim_{x\to 1}f(x)= 4$ and $\displaystyle \lim_{x\to 0}4\cdot \frac{x}{x}$ is undefined since $\displaystyle \lim_{x\to 0^+}4\cdot \frac{x}{x}=4$ and $\displaystyle \lim_{x\to 0^}4\cdot \frac{x}{x}= 4$
Is this the kind of counterexample you're looking for ?

thanks for your help.
I thought that f(x)=4x is an example for a function that do not follow the rules the question specified. however, when I tried to answer the question using f(x) = 4x, the computer application claimed it is not the correct example.
I still think that f(x) = 4x satisfies my condition, thank for your help. let me know if you have another function that do not follow the rules.
as for the first problem, it is correct...and thank you for your help.