Answer
a) 0; b) 0;
Work Step by Step
We are given the function:
$f(x)=\dfrac{x-2}{\ln |x-2|}$
a) Graph $f$ to estimate $\lim\limits_{x \to 2} f(x)$:
From the graph we find:
$\lim\limits_{x \to 2} f(x)=0$
b) Evaluate $f(x)$ for values near 2:
$f(1.9)\approx 0.04343$
$f(1.99)\approx 0.00217$
$f(1.999)\approx 0.000145$
$f(2.001)\approx -0.000148$
$f(2.01)\approx -0.00217$
$f(2.1)\approx -0.0434$
The values we found prove that the conjecture from part $a)$ is true:
$\lim\limits_{x \to 2} f(x)=0$