Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 36

(a) Illustrate the function $\displaystyle y = \frac{\tan 4x}{x}$ by graphing and zooming in toward the point where the graph crosses the $y$-axis, estimate the value of $\lim \limits_{x \to 0} f(x)$.



When the viewing rectangle is a small number, the limit of the function is identifiable at 4 as shown on the graph on the left side.
However, if the viewing rectangle approaches a large number, the limit of the function will not be clearly identifiable.

(b) Evaluate $f(x)$ for values of $x$ that approach 0. Check if the results match your answer in part($x$).




$
\begin{array}{|c|c|}
\hline
x & f(x)\displaystyle \frac{\tan 4x}{x}\\
\hline
0.01 & 4.0021347\\
0.001 & 4.0000213\\
0.0001 & 4.0000002\\
-0.01 & 4.0021347\\
-0.001 & 4.0000213\\
-0.0001 & 4.0000002\\
\hline
\end{array}
$



$
\begin{equation}
\begin{aligned}
\lim\limits_{x \to 0} \displaystyle \frac{\tan 4x}{x} & = \frac{\tan [4(0.0001)]}{0.0001 } = 4.0000002\\
\lim\limits_{x \to 0} \displaystyle \frac{\tan 4x}{x} & = 4.0000002
\end{aligned}
\end{equation}
$