Provide an appropriate response.
-It can be shown that the inequalities $- x \leq x \cos \left( \frac { 1 } { x } \right) \leq x$ hold for all values of $x \geq 0$. Find $\lim _ { x \rightarrow 0 } x \cos \left( \frac { 1 } { x } \right)$ if it exists.
A) 1
B) 0
C) $0.0007$
D) does not exist

Question

Provide an appropriate response.
-It can be shown that the inequalities $- x \leq x \cos \left( \frac { 1 } { x } \right) \leq x$ hold for all values of $x \geq 0$. Find $\lim _ { x \rightarrow  0 } x \cos \left( \frac { 1 } { x } \right)$ if it exists.
A) 1
B) 0
C) $0.0007$
D) does not exist

Quizplus · Accepted Answer

From the given inequalities, we have:
$-x \leq x \cos\left(\frac{1}{x}\right) \leq x$
Dividing all sides by $x$ (which is positive since we are considering values of $x$ greater than or equal to 0), we get:
$-1 \leq \cos\left(\frac{1}{x}\right) \leq 1$
Since $-1 \leq \cos\left(\frac{1}{x}\right) \leq 1$ for all values of $x \geq 0$, we can apply the squeeze theorem to find the limit as $x$ approaches 0:
$$\lim_{x\rightarrow 0} \cos\left(\frac{1}{x}\right) = 1$$
Thus,
$$\lim_{x\rightarrow 0} x \cos\left(\frac{1}{x}\right) = 0*1 = 0$$
Therefore, the best choice is B) 0.

Provide an Appropriate Response −x≤xcos⁡(1x)≤x- x \leq x \cos \left( \frac { 1 } { x } \right) \leq x−x≤xcos(x1​)≤x

Provide an Appropriate Response $- x \leq x \cos \left( \frac { 1 } { x } \right) \leq x$