Textbook Question
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx → √3 1/x² = 1/3
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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
Verified step by step guidance1
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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Video duration:
4mKey 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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One-Sided Limits
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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05:44Derivatives
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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05:13Slopes of Tangent Lines
Related Practice
Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limx→1− (1/(x + 1))((x + 6)/x)((3 − x)/7)
Textbook Question
Calculating Limits
Find the limits in Exercises 11–22.
limx→−1/2 4x(3x+4)²
Textbook Question
Limits of quotients
Find the limits in Exercises 23–42.
limx→−1 (√(x² + 8) − 3) / (x + 1)
Textbook Question
Limits and Infinity
Find the limits in Exercises 37–46.
x²/³ + x⁻¹
lim --------------------
x→∞ x²/³ + cos²x
Textbook Question
Finding Limits
In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)
f(x) = 2/x − 3