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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 89
Chapter 2, Problem 89

Suppose g(x) = {x^2−5x if x≤−1
ax^3−7 if x>−1.
Determine a value of the constant a for which lim x→−1 g(x) exists and state the value of the limit, if possible.

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1
Identify the piecewise function g(x) and the point of interest x = -1.
For the limit \( \lim_{x \to -1} g(x) \) to exist, the left-hand limit and the right-hand limit as x approaches -1 must be equal.
Calculate the left-hand limit: \( \lim_{x \to -1^-} g(x) = \lim_{x \to -1^-} (x^2 - 5x) \).
Calculate the right-hand limit: \( \lim_{x \to -1^+} g(x) = \lim_{x \to -1^+} (ax^3 - 7) \).
Set the left-hand limit equal to the right-hand limit and solve for the constant a.

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Key Concepts

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

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. In this case, we need to evaluate the limit of g(x) as x approaches -1 from both sides. For the limit to exist, the left-hand limit and the right-hand limit must be equal.
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Limits of Rational Functions: Denominator = 0

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this problem, g(x) has two different expressions depending on whether x is less than or equal to -1 or greater than -1. Understanding how to evaluate each piece is crucial for finding the limit at the point where the definition changes.
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Piecewise Functions

Continuity at a Point

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. For the limit of g(x) to exist at x = -1, we must find a value of 'a' such that the left-hand limit (from x ≤ -1) equals the right-hand limit (from x > -1) and also equals g(-1).
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Intro to Continuity
Related Practice
Textbook Question

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 a.m. on Friday morning, and let g(t) be your distance from the car t hours after 7 a.m. on Sunday morning.


b. Let h(t)=f(t)−g(t). Find h(0) and h(2).

Textbook Question

Use an appropriate limit definition to prove the following limits.


lim x→1 (5x−2) =3;

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Textbook Question

Suppose f(x) = {x^2 − 5x + 6 / x − 3 if x≠3

a if x=3.

Determine a value of the constant a for which lim x→3 f(x) = f(3).

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Textbook Question

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 a.m. on Friday morning, and let g(t) be your distance from the car t hours after 7 a.m. on Sunday morning.


a. Evaluate f(0), f(2), g(0), and g(2).

Textbook Question

A sine limit It can be shown that 1−x^2/ 6 ≤ sin x/ x ≤1, for x near 0.

Use these inequalities to evaluate lim x→0 sin x/ x.

Textbook Question

Calculate the following limits using the factorization formula x^n−a^n=(x−a)(x^n−1+ax^n−2+a^2x^n−3+⋯+a^n−2x+a^n−1), where n is a positive integer and a is a real number.

lim x→1 x^6 − 1 / x − 1

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