To solve this problem, we need to apply the Squeeze Theorem.

Given:

We are asked to find:

$\lim_{x \to 7} f(x)$

with the inequality:

$14x - 49 \leq f(x) \leq x^2 \quad \text{on} \quad [5, 9].$

Squeeze Theorem:

If a function $f(x)$ is "squeezed" between two functions $g(x)$ and $h(x)$ near a point $a$, and if both $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then:

$\lim_{x \to a} f(x) = L.$

Step-by-Step Solution:

1. The bounds are: $14x - 49 \leq f(x) \leq x^2$

2. Let's find the limits of the bounding functions as $x \to 7$.

   • For the lower bound $g(x) = 14x - 49$: $\lim_{x \to 7} (14x - 49) = 14(7) - 49 = 98 - 49 = 49.$

   • For the upper bound $h(x) = x^2$: $\lim_{x \to 7} x^2 = 7^2 = 49.$

3. Since both the lower and upper bounds approach 49 as $x \to 7$, by the Squeeze Theorem, we conclude: $\lim_{x \to 7} f(x) = 49.$

Answer:

$\text{Limit} = 49$

Would you like more details on any step? Also, here are some related questions:

1. What are the conditions for using the Squeeze Theorem?
2. How do you find limits of polynomial functions?
3. Can the Squeeze Theorem be applied when one of the bounding functions doesn't exist at a certain point?
4. What happens if the two bounding functions have different limits?
5. How do we apply the Squeeze Theorem if the functions are not continuous?

Tip: Always check that the bounding functions have the same limit at the point where you're applying the Squeeze Theorem!