Answer
(a) (i) $\lim\limits_{x \to 0^+}f(x) = 0$
(ii) $\lim\limits_{x \to 1^-}f(x) = -\infty$
(iii) $\lim\limits_{x \to 1^-}f(x) = \infty$
(b) Since the values seem to increase without bound, we could guess that:
$\lim\limits_{x \to \infty}f(x) = \infty$
(c) We can see a sketch of the graph below.
Work Step by Step
(a) $f(x) = \frac{x}{ln~x}$
(i) As $~~x \to 0^+~~$, the value of $x$ is a very small positive number, while $ln~x$ approaches $-\infty$
$\lim\limits_{x \to 0^+}f(x) = 0$
(ii) As $~~x \to 1^-~~$, the value of $x$ is just less than 1, while $ln~x$ is a very small negative number.
$\lim\limits_{x \to 1^-}f(x) = -\infty$
(iii) As $~~x \to 1^+~~$, the value of $x$ is a little more than 1, while $ln~x$ is a very small positive number.
$\lim\limits_{x \to 1^-}f(x) = \infty$
(b) We can evaluate $f(x)$ for increasing values of $x$:
$f(10) = \frac{10}{ln~10} = 4.34$
$f(100) = \frac{100}{ln~100} = 21.7$
$f(1000) = \frac{1000}{ln~1000} = 144.8$
$f(10,000) = \frac{10,000}{ln~10,000} = 1085.7$
$f(100,000) = \frac{100,000}{ln~100,000} = 8685.9$
$f(1,000,000) = \frac{1,000,000}{ln~1,000,000} = 72,382$
Since the values seem to increase without bound, we could guess that:
$\lim\limits_{x \to \infty}f(x) = \infty$
(c) We can see a sketch of the graph below.
