Provide an appropriate response.
-If $x ^ { 3 } \leq f ( x ) \leq x$ for $x$ in $[ - 1,1 ]$, find $\lim _ { x \rightarrow \theta } f ( x )$ if it exists.
A) does not exist
B) 1
C) $- 1$
D) 0

Question

Quizplus · Accepted Answer

Since $x^3 \leq f(x) \leq x$ for all $x$ in $[-1,1]$, we have 
$$-x\leq -f(x)\leq -x^3$$ 
$$\Rightarrow x^3\leq x-f(x)\leq x$$ 
Now taking limits as $x$ approaches 0 using squeeze theorem, we get 
$$\lim _ { x \rightarrow 0 } x ^ { 3 } \leq \lim _ { x \rightarrow 0 } (x - f ( x )) \leq\lim _ { x \rightarrow 0 } x$$ 
$$0\leq \lim _ { x \rightarrow 0 } (x - f ( x )) \leq 0$$ 
Hence the limit exists and is equal to 0.

Provide an Appropriate Response x3≤f(x)≤xx ^ { 3 } \leq f ( x ) \leq xx3≤f(x)≤x

Provide an Appropriate Response $x ^ { 3 } \leq f ( x ) \leq x$