Question

Suppose -2x² - 20x - 51 ≤ f(x) ≤ 2x² + 20x + 49 for all x values near -5, except at -5. Evaluate lim x→-5 f(x).

a) 324
b) 625
c) -324
d) -625

Answer 1

The limit of f(x) as x approaches -5 is -625.

Step-by-step explanation:

To evaluate the limit lim x→-5 f(x), we need to find the value of f(x) as x approaches -5. Given the inequality -2x² - 20x - 51 ≤ f(x) ≤ 2x² + 20x + 49, we can see that the lower bound function is concave down and the upper bound function is concave up. Both functions have the same leading coefficient, so the value of f(x) will be sandwiched between them as x approaches -5.

By substituting -5 into the lower bound function, we get -2(-5)² - 20(-5) - 51 = -324. Likewise, substituting -5 into the upper bound function, we get 2(-5)² + 20(-5) + 49 = -625. Since the function f(x) is bounded between these two values, the limit lim x→-5 f(x) is equal to -625.