Answer
$\textbf{(a)} 0$, $\textbf{(b)}+\infty$, $\textbf{(c)} 0$, $\textbf{(d)}-\infty$, $\textbf{(e)}$ $+\infty$, $\textbf{(f)}$ $-\infty$
Work Step by Step
$\textbf{(a)}$
$$\lim\limits_{x \to 0^{+}}{\frac{x}{\ln x}} = \lim\limits_{x \to 0^{+}}{x} \cdot \lim\limits_{x \to 0^{+}}{\frac{1}{\ln x}} = 0 \cdot \frac{1}{-\infty} = \frac{0}{-\infty} = 0$$
$\textbf{(b)}$
$$\lim\limits_{x \to +\infty}{\frac{x^3}{e^{-x}}} =\lim\limits_{x \to +\infty}{\frac{x^3}{1/e^x}} =\lim\limits_{x \to +\infty}{e^xx^3} =\lim\limits_{x \to +\infty}{e^x} \cdot \lim\limits_{x \to +\infty}{x^3} = +\infty\cdot+\infty = +\infty$$
$\textbf{(c)}$
$$\lim\limits_{x \to (\pi/2)^-}(\cos x)^{\tan x} = \lim\limits_{x \to (\pi/2)^-}(\cos x)^{\lim\limits_{x \to (\pi/2)^-}\tan x} = 0^{+\infty} = 0$$
$\textbf{(d)}$
$$\lim\limits_{x \to 0^+}(\ln x)\cot x =\lim\limits_{x \to 0^+}(\ln x) \cdot \lim\limits_{x \to 0^+}(\cot x) = (-\infty) \cdot (+\infty) = -\infty $$
$\textbf{(e)}$
$$\lim\limits_{x \to 0^+}\left(\frac{1}{x} - \ln x\right) = \lim\limits_{x \to 0^+}\left(\frac{1}{x}\right) - \lim\limits_{x \to 0^+}(\ln x) = +\infty - (-\infty) = +\infty$$
$\textbf{(f)}$
$$\lim\limits_{x \to 0^+}(x - x^3) = \lim\limits_{x \to 0^+}x - \lim\limits_{x \to 0^+}x^3 = -\infty + (-\infty) = -\infty$$
