Limits Properties — Question 10

Question 10 - Solution

We are told that for all $x$ near 2: $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x \to 2} g(x) = 4, \quad \lim_{x \to 2} h(x) = 4$$

This is a perfect scenario to apply the "Squeeze Theorem", which states:

If $g(x) \leq f(x) \leq h(x)$ near $x = a$, and $$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$$ then $$\lim_{x \to a} f(x) = L$$

Applying this here: $$\lim_{x \to 2} f(x) = \boxed{4}$$

Conclusion: By the Squeeze Theorem, since both bounding functions tend to 4 as $x \to 2$, so must $f(x)$.