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

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

Question

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

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

Quizplus · Accepted Answer

When we plug in $x=0$, we get $\frac{\sin(0)}{0}$, which is undefined. This may indicate that the limit does not exist, but we need to check further.

Notice that as $x$ approaches $0$ from the left or right, the function oscillates rapidly between $-8$ and $8$. However, the numerator will always be squeezed between $-1$ and $1$, so the fraction is always squeezed between $\frac{-1}{x}$ and $\frac{1}{x}$.

As $x$ gets very small (but never equal to $0$), $\frac{-1}{x}$ and $\frac{1}{x}$ both tend towards infinity (one negative, one positive). Since our function is always squeezed between these two, it must also tend towards infinity (or negative infinity) as $x	o0$.

Since the limit is not a real number, but rather $\pm\infty$, the answer is $\boxed{	ext{(E) The limit does not exist.}}$

Consider the Graph of the Function and Approximate lim⁡x→0sin⁡(8x)x\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }limx→0​xsin(8x)​

Consider the Graph of the Function and Approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( 8 x ) } { x }$