Answer
See explanations.
Work Step by Step
(a) Assume $$\lim_{x\to\infty}f(x)=L$$
Let $t=\frac{1}{x}$, we have: $$\lim_{x\to\infty}t=0^+$$
Replace the variable $x$ with $t$ in the first limit, we have: $$\lim_{t\to 0^+}f(\frac{1}{t})=L$$
Thus we have: $$\lim_{x\to\infty}f(x)=\lim_{t\to 0^+}f(\frac{1}{t})$$
(b) Similarly, assume $$\lim_{x\to -\infty}f(x)=K$$
Let $t=\frac{1}{x}$, we have: $$\lim_{x\to -\infty}t=0^-$$
Replace the variable $x$ with $t$ in the starting limit, we have: $$\lim_{t\to 0^-}f(\frac{1}{t})=K$$
Thus we have: $$\lim_{x\to -\infty}f(x)=\lim_{t\to 0^-}f(\frac{1}{t})$$