Answer
a) 2; b) 2;
Work Step by Step
We are given the function:
$f(x)=\dfrac{1-\cos(2x-2)}{(x-1)^2}$
a) Graph $f$ to estimate $\lim\limits_{x \to 1} f(x)$:
From the graph we find:
$\lim\limits_{x \to 1} f(x)=2$
b) Evaluate $f(x)$ for values near 1:
$f(0.9)\approx 1.9933422$
$f(0.99)\approx 1.9999333$
$f(0.999)\approx 1.9999993$
$f(1.001)\approx 1.9999993$
$f(1.01)\approx 1.9999333$
$f(1.1)\approx 1.9933422$
The values we found prove that the conjecture from part $a)$ is true:
$\lim\limits_{x \to 1} f(x)=2$
