Answer
(a) It appears that $\lim\limits_{x \to 0}\frac{tan~4x}{x} = 4$
(b) We can see that $\lim\limits_{x \to 0}\frac{tan~4x}{x} = 4$
Work Step by Step
(a) By zooming in on the point where the function $f(x) = \frac{tan~4x}{x}$ crosses the y-axis, it appears that $\lim\limits_{x \to 0}\frac{tan~4x}{x} = 4$
(b) We can evaluate $f(x)$ for values of $x$ that approach $0$:
$f(0.1) = \frac{tan~[4(0.1)]}{0.1} = 4.228$
$f(0.01) = \frac{tan~[4(0.01)]}{0.01} = 4.002$
$f(0.001) = \frac{tan~[4(0.001)]}{0.001} = 4.00002$
$f(0.0001) = \frac{tan~[4(0.0001)]}{0.0001} = 4.0000002$
$f(-0.1) = \frac{tan~[4(-0.1)]}{-0.1} = 4.228$
$f(-0.01) = \frac{tan~[4(-0.01)]}{-0.01} = 4.002$
$f(-0.001) = \frac{tan~[4(-0.001)]}{-0.001} = 4.00002$
$f(-0.0001) = \frac{tan~[4(-0.0001)]}{-0.0001} = 4.0000002$
We can see that $\lim\limits_{x \to 0}\frac{tan~4x}{x} = 4$
