Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = x², x = -2

Verified step by step guidance

1. Identify the function f(x) = x² and the point x = -2 where we need to evaluate the limit.
2. Substitute f(x) = x² into the limit expression: lim(h→0) [(f(x+h) - f(x)) / h]. This becomes lim(h→0) [((x+h)² - x²) / h].
3. Expand the expression (x+h)² to get x² + 2xh + h². Substitute this back into the limit expression: lim(h→0) [(x² + 2xh + h² - x²) / h].
4. Simplify the expression by canceling out x²: lim(h→0) [(2xh + h²) / h].
5. Factor out h from the numerator: lim(h→0) [h(2x + h) / h]. Cancel h from the numerator and denominator, resulting in lim(h→0) [2x + h]. Now, substitute x = -2 and evaluate the limit as h approaches 0.

Key Concepts

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. It is fundamental for understanding continuity, derivatives, and integrals. The notation lim h→0 (f(x+h) - f(x)) / h specifically represents the limit of the average rate of change of a function as the interval approaches zero, which leads to the concept of the derivative.

Derivatives

The derivative of a function at a point quantifies the instantaneous rate of change of the function with respect to its variable. It is defined as the limit of the average rate of change as the interval shrinks to zero. In the context of the given question, finding the derivative of f(x) = x² at x = -2 involves evaluating the limit expression provided.

Secant and Tangent Lines

Secant lines connect two points on a curve and represent the average rate of change between those points. As the two points get infinitely close, the secant line approaches the tangent line, which touches the curve at a single point and represents the instantaneous rate of change. Understanding this relationship is crucial for grasping how limits lead to derivatives in calculus.