Answer
a) 3; b) 3;
Work Step by Step
We are given the function:
$g(x)=\dfrac{3\sin x-2\cos x+2}{x}$
a) Graph $g$ to estimate $\lim\limits_{x \to 0} g(x)$:
From the graph we find:
$\lim\limits_{x \to 0} g(x)=3$
b) Evaluate $g(x)$ for values near 0:
$g(-0.1)\approx 2.8950858$
$g(-0.01)\approx 2.9899501$
$g(-0.001)\approx 2.9989995$
$g(0.001)\approx 3.0009995$
$g(0.01)\approx 3.0099499$
$g(0.1)\approx 3.0949192$
The values we found prove that the conjecture from part $a)$ is true:
$\lim\limits_{x \to 0} g(x)=3$
