Question

If f is continuous at 5 and f(5) = 2 and f(4) = 3 then what is the lim(x --> 2) f(4x² - 11) equal?

Answer 1

The limit (x → 2) f(4x² - 11) is equal to 2 because the function f is continuous at 5, and as x approaches 2, the inner function 4x² - 11 approaches 5, where f(5) is given as 2.

Step-by-step explanation:

To solve the limit (x → 2) f(4x² - 11), we first need to understand the behavior of the inner function 4x² - 11 as x approaches 2. We find that:

1. 4x² - 11 = 4(2)² - 11 = 4(4) - 11 = 16 - 11 = 5

Since f is continuous at 5 and f(5) = 2, as stated in the question, the value of the function f at 5 is 2. Therefore, as x approaches 2, the inner function 4x² - 11 approaches 5.

So, the limit can be directly evaluated as:

L(x → 2) f(4x² - 11) = f(5) = 2.

The reason we can make this evaluation directly is that continuity of f at a point guarantees that the limit of f as x approaches that point is equal to the function value at that point.