Limit of one function given limits of another

Question

If a certain function $f$ is known to have the properties

$$ \lim_{x \to -\infty} f(x) = 4 \qquad \text{and} \qquad \lim_{x \to \infty} f(x) = 6, $$

how would I determine if

$$ \lim_{x \to 0^+} \frac{1+3x}{3+f(\frac{1}{x})} $$

exists, and if it does, compute its value?

Answer 1

you have that $\lim_{x\to\infty}{f(x)}=6 \Rightarrow\lim_{u\to0^+}{f(\frac{1}{u})}=6.$So the limit that you want is $\frac{1}{9}$

Answer 2

Set $y:= 1/x;$

$\lim_{ y \rightarrow +\infty} \dfrac{1+3/y}{3+f(y)}.$

Can you take it from here?