Provide an appropriate response.
-The inequality $1 - \frac { x ^ { 2 } } { 2 }

Question

Provide an appropriate response.
-The inequality $1 - \frac { x ^ { 2 } } { 2 } < \frac { \sin x } { x } < 1$ holds when $x$ is measured in radians and $| x | < 1$. Find $\lim _ { x \rightarrow \theta } \frac { \sin x } { x }$ if it exists.
A) 0
B) 1
C) does not exist
D) $0.0007$

Quizplus · Accepted Answer

We know that $\frac {\sin x}{x}$ is sandwiched between $1-\frac{x^2}{2}$ and $1$. As $x$ approaches $0$, both of these bounds approach $1$. Therefore, by the squeeze theorem, the limit of $\frac {\sin x}{x}$ as $x$ approaches $0$ is $\boxed{\textbf{(B) }1}$.

Provide an Appropriate Response 1−x22<sin⁡xx<11 - \frac { x ^ { 2 } } { 2 } < \frac { \sin x } { x } < 11−2x2​<xsinx​<1

Provide an Appropriate Response $1 - \frac { x ^ { 2 } } { 2 } < \frac { \sin x } { x } < 1$