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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 2.2.80a
Chapter 2, Problem 2.2.80a

Theory and Examples


a. If limx→0 f(x) / x² = 1, find limx→0 f(x).

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First, understand the given limit: \( \lim_{x \to 0} \frac{f(x)}{x^2} = 1 \). This means that as \( x \) approaches 0, the function \( \frac{f(x)}{x^2} \) approaches 1.
To find \( \lim_{x \to 0} f(x) \), consider the behavior of \( f(x) \) as \( x \) approaches 0. Since \( \frac{f(x)}{x^2} \to 1 \), it implies that \( f(x) \) behaves like \( x^2 \) near 0.
Multiply both sides of the equation \( \frac{f(x)}{x^2} = 1 \) by \( x^2 \) to isolate \( f(x) \): \( f(x) = x^2 \cdot 1 = x^2 \).
Now, substitute \( f(x) = x^2 \) into the limit expression: \( \lim_{x \to 0} f(x) = \lim_{x \to 0} x^2 \).
Evaluate the limit \( \lim_{x \to 0} x^2 \). As \( x \to 0 \), \( x^2 \to 0 \). Therefore, \( \lim_{x \to 0} f(x) = 0 \).

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of f(x) as x approaches 0. Understanding limits is crucial for analyzing the continuity and behavior of functions at specific points.
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One-Sided Limits

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can simplify the process of finding limits in complex scenarios.
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Power Rules

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In this problem, if we find that limx→0 f(x) exists and equals a specific value, it indicates that f(x) is continuous at x = 0, which is essential for understanding the behavior of the function around that point.
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Related Practice
Textbook Question

Finding One-Sided Limits Algebraically


Find the limits in Exercises 11–20.


a. limx→0+ (1 − cos x) / |cos x − 1|

Textbook Question

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


h(t)=cot t


a. [π/4,3π/4]

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


a. ƒ(x) = x¹/³

Textbook Question

Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let g(x) = (x² − 2) / (x − √2)


a. Make a table of the values of g at the points x=1.4,1.41,1.414, and so on through successive decimal approximations of √2. Estimate limx→√2 g(x).

Textbook Question

Exercises 5–10 refer to the function

f(x) = { x² − 1, −1 ≤ x < 0

2x, 0 < x < 1

1, x = 1

−2x + 4, 1 < x < 2

0, 2 < x < 3

graphed in the accompanying figure.

<IMAGE>

a. Does f (1) exist?

Textbook Question

Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find


a. limx→−2 (p(x) + r(x) + s(x))