Questions: Use the Squeeze Theorem to find the limit as x approaches 7 of f(x), given that 6 - x - 7 ≤ f(x) ≤ 6 + x - 7.

Transcript text: Use the Squeeze Theorem to find $\lim _{x \rightarrow 7} f(x)$, given that $6-|x-7| \leq f(x) \leq 6+|x-7|$.

Solution

Solution Steps

To find the limit of $ f(x) $ as $ x $ approaches 7 using the Squeeze Theorem, we need to evaluate the limits of the bounding functions $ 6 - |x-7| $ and $ 6 + |x-7| $ as $ x $ approaches 7. If both limits are equal, then the limit of $ f(x) $ will also be equal to this common value.

Step 1: Identify the Bounds

We are given the inequalities $ 6 - |x - 7| \leq f(x) \leq 6 + |x - 7| $. These inequalities provide the lower and upper bounds for the function $ f(x) $.

Step 2: Evaluate the Limits of the Bounds

Next, we evaluate the limits of the lower and upper bounds as $ x $ approaches 7: \[ \lim_{x \to 7} (6 - |x - 7|) = 6 - |7 - 7| = 6 - 0 = 6 \] \[ \lim_{x \to 7} (6 + |x - 7|) = 6 + |7 - 7| = 6 + 0 = 6 \]

Step 3: Apply the Squeeze Theorem

Since both the lower and upper limits as $ x $ approaches 7 are equal to 6, by the Squeeze Theorem, we conclude that: \[ \lim_{x \to 7} f(x) = 6 \]