Questions: If lim x→2 f(x)=lim x→2 h(x)=11, find lim x→2 g(x) when f(x) ≤ g(x) ≤ h(x) for all values of x.

Transcript text: If $\lim _{x \rightarrow 2} f(x)=\lim _{x \rightarrow 2} h(x)=11$, find $\lim _{x \rightarrow 2} g(x)$ when $f(x) \leq g(x) \leq h(x)$ for all values of $x$.

Solution

Solution Steps

To solve this problem, we can use the Squeeze Theorem (also known as the Sandwich Theorem). According to this theorem, if $ f(x) \leq g(x) \leq h(x) $ for all $ x $ in some interval around $ a $ (except possibly at $ a $), and if $ \lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L $, then $ \lim_{x \to a} g(x) = L $. In this case, since both $ \lim_{x \to 2} f(x) $ and $ \lim_{x \to 2} h(x) $ equal 11, and $ f(x) \leq g(x) \leq h(x) $, it follows that $ \lim_{x \to 2} g(x) = 11 $.

Step 1: Identify the Given Limits

We are given that $\lim_{x \to 2} f(x) = 11$ and $\lim_{x \to 2} h(x) = 11$.

Step 2: Apply the Squeeze Theorem

The Squeeze Theorem states that if $f(x) \leq g(x) \leq h(x)$ for all $x$ in some interval around 2, and both $\lim_{x \to 2} f(x)$ and $\lim_{x \to 2} h(x)$ equal the same value, then $\lim_{x \to 2} g(x)$ must also equal that value.

Step 3: Conclude the Limit of $g(x)$

Since both $\lim_{x \to 2} f(x)$ and $\lim_{x \to 2} h(x)$ are 11, it follows from the Squeeze Theorem that $\lim_{x \to 2} g(x) = 11$.