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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.2.60
Chapter 2, Problem 2.2.60

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) = 1/x, x = -2

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1
Identify the function f(x) = 1/x and the point x = -2 where we need to evaluate the limit.
Set up the expression for the average rate of change: (f(x+h) - f(x)) / h. Substitute f(x) = 1/x into this expression.
Substitute x = -2 into the expression, resulting in (f(-2+h) - f(-2)) / h.
Calculate f(-2+h) = 1/(-2+h) and f(-2) = 1/(-2). Substitute these into the expression to get: (1/(-2+h) - 1/(-2)) / h.
Simplify the expression: Combine the fractions in the numerator to get a single fraction, and then simplify the entire expression. Finally, take the limit as h approaches 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Derivatives

The derivative of a function at a point quantifies the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval shrinks to zero. Derivatives are essential for analyzing the behavior of functions, including finding slopes of tangent lines and optimizing functions.
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Secant and Tangent Lines

A secant line intersects a curve at two or more points, representing the average rate of change between those points. In contrast, a tangent line touches the curve at a single point and represents the instantaneous rate of change at that point. The transition from secant to tangent lines is crucial in calculus, as it leads to the definition of derivatives and the concept of limits.
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Related Practice
Textbook Question

Limits of quotients


Find the limits in Exercises 23–42.


limx→−1 (√(x² + 8) − 3) / (x + 1)

Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


lim x→1 1/x = 1

Textbook Question

At what points are the functions in Exercises 13–30 continuous?


g(x) = { (x² − x – 6)/(x – 3), x ≠ 3

5, x = 3


Textbook Question

Suppose that f(x) and g(x) are polynomials in x. Can the graph of f(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.

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Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x → ∞ √((8x² − 3) / (2x² + x))

Textbook Question

Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.