Question

Evaluate the limit as h approaches 0 of (f(2+h) - f(2)) / h, where f(x) = -3x² + 2. If the limit does not exist, enter DNE.

a. -12
b. -6
c. 0
d. DNE

Answer 1

The limit as h approaches 0 of the expression (f(2+h) - f(2)) / h for the function f(x) = -3x² + 2 is -12. This corresponds to the derivative of f at x=2.

Step-by-step explanation:

The question asks to evaluate the limit of the expression (f(2+h) - f(2)) / h as h approaches 0, where f(x) is defined as -3x² + 2. This is a typical problem of finding the derivative of a function at a given point using the definition of the derivative.

First, we substitute f(x) into the expression:

(f(2+h) - f(2)) / h = (-3(2+h)² + 2 - (-3(2)² + 2)) / h

Now, expanding the square and simplifying:

= (-3(4+4h+h²) + 2 - (-12 + 2)) / h

= (-12 - 12h - 3h² + 2 + 12 - 2) / h

= (-12h - 3h²) / h

Canceling h from the numerator and denominator:

= -12 - 3h

As h approaches 0, the term -3h vanishes, and we are left with:

-12

Thus, the limit as h approaches 0 of the given expression is -12, which corresponds to choice (a).