33. if 2 ≤ f(x) ≤ x² + 2 for all x, find lim(x→0) f(x). sketch a graph to illustrate.

Explanation:

Step1: Find left - hand limit

We know that $\lim_{x\rightarrow0}2 = 2$.

Step2: Find right - hand limit

Calculate $\lim_{x\rightarrow0}(x^{2}+2)$. Substitute $x = 0$ into $x^{2}+2$, we get $\lim_{x\rightarrow0}(x^{2}+2)=0^{2}+2 = 2$.

Step3: Apply Squeeze Theorem

Since $2\leq f(x)\leq x^{2}+2$ for all $x$ and $\lim_{x\rightarrow0}2=\lim_{x\rightarrow0}(x^{2}+2) = 2$, by the Squeeze Theorem, $\lim_{x\rightarrow0}f(x)=2$.

Answer:

$2$