Answer
(a) $\lim\limits_{x \to \infty}f(x) = 0$
(b) $\lim\limits_{x \to 0^+}f(x) = \infty$
(c) $\lim\limits_{x \to 1^-}f(x) = \infty$
(d) $\lim\limits_{x \to 1^-}f(x) = -\infty$
(e) We can see a sketch of the graph below.
Work Step by Step
$f(x) = \frac{2}{x} - \frac{1}{ln~x}$
(a) As $~~x \to \infty~~$, the value of $\frac{2}{x}$ approaches $0$ while $\frac{1}{ln~x}$ approaches $0$
$\lim\limits_{x \to \infty}f(x) = 0$
(b) As $~~x \to 0^+~~$, the value of $\frac{2}{x}$ approaches $\infty$ while $\frac{1}{ln~x}$ approaches $0$
$\lim\limits_{x \to 0^+}f(x) = \infty$
(c) As $~~x \to 1^-~~$, the value of $\frac{2}{x}$ approaches $2$ while $\frac{1}{ln~x}$ approaches $-\infty$
$\lim\limits_{x \to 1^-}f(x) = \infty$
(d) As $~~x \to 1^+~~$, the value of $\frac{2}{x}$ approaches $2$ while $\frac{1}{ln~x}$ approaches $\infty$
$\lim\limits_{x \to 1^-}f(x) = -\infty$
(e) We can see a sketch of the graph below.
