Answer
(a) When we zoom in toward the point where the graph crosses the y-axis, we can estimate that $\lim\limits_{x \to 0}f(x) = -1.5$
(b) We can see that $\lim\limits_{x \to 0}f(x) = -1.5$
Work Step by Step
(a) When we zoom in toward the point where the graph crosses the y-axis, we can estimate that $\lim\limits_{x \to 0}f(x) = -1.5$
(b) We can evaluate $f(x)$ for values of $x$ that approach $0$:
$f(0.1) = \frac{cos [2(0.1)]-cos(0.1)}{(0.1)^2} = -1.493759$
$f(-0.1) = \frac{cos [2(-0.1)]-cos(-0.1)}{(-0.1)^2} = -1.493759$
$f(0.01) = \frac{cos [2(0.01)]-cos(0.01)}{(0.01)^2} = -1.49994$
$f(-0.01) = \frac{cos [2(-0.01)]-cos(-0.01)}{(-0.01)^2} = -1.49994$
$f(0.001) = \frac{cos [2(0.001)]-cos(0.001)}{(0.001)^2} = -1.499999$
$f(-0.001) = \frac{cos [2(-0.001)]-cos(-0.001)}{(-0.001)^2} = -1.499999$
$f(0.0001) = \frac{cos [2(0.0001)]-cos(0.0001)}{(0.0001)^2} = -1.5$
$f(-0.0001) = \frac{cos [2(-0.0001)]-cos(-0.0001)}{(-0.0001)^2} = -1.5$
We can see that $\lim\limits_{x \to 0}f(x) = -1.5$
