$$\text { Consider the graph of the function and approximate } \lim _ { x \rightarrow 0 } \frac { \sin ( - 8 x ) } { x } \text {, if it exists. }$$

A) -1
B) -4
C) -8
D) 1
E) The limit does not exist.

Question

$$\text { Consider the graph of the function and approximate } \lim _ { x \rightarrow 0 } \frac { \sin ( - 8 x ) } { x } \text {, if it exists. }$$

A) -1 
B) -4 
C) -8 
D) 1 
E) The limit does not exist.

Quizplus · Accepted Answer

As $x$ approaches 0, $-8x$ approaches 0 as well. Therefore, we can use the property that $\lim_{u\rightarrow 0}\frac{\sin u}{u}=1$, where $u=-8x$. Thus, $\lim_{x\rightarrow 0}\frac{\sin(-8x)}{x}=\lim_{u\rightarrow 0}\frac{\sin u}{-8u}=-\frac{1}{8}\cdot 1=-\frac{1}{8}$. Therefore, the best choice is (C).