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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 87
Chapter 2, Problem 87

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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1
Identify that the problem requires finding the limit of f(x) as x approaches 3 and ensuring it equals f(3).
Recognize that f(x) is a piecewise function, and for x ≠ 3, f(x) = \( \frac{x^2 - 5x + 6}{x - 3} \).
Factor the numerator \( x^2 - 5x + 6 \) to simplify the expression: \( (x - 2)(x - 3) \).
Cancel the common factor \( (x - 3) \) in the numerator and denominator, simplifying f(x) to \( x - 2 \) for x ≠ 3.
Evaluate the limit \( \lim_{{x \to 3}} (x - 2) \) and set it equal to a, ensuring continuity at x = 3.

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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 3. For the limit to exist, the values of f(x) must approach a specific number as x gets closer to 3 from both sides.
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One-Sided Limits

Continuity

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 scenario, we need to find a value for the constant 'a' such that the limit of f(x) as x approaches 3 equals f(3), ensuring that the function is continuous at x = 3.
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Intro to Continuity

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this problem, f(x) is defined differently for x = 3 compared to x ≠ 3. Understanding how to evaluate and manipulate piecewise functions is crucial for determining the appropriate value of 'a' that maintains the function's continuity at x = 3.
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Related Practice
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 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. Use the Intermediate Value Theorem to show that the equation has a solution in the given interval.


x=cos x; (0,π/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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Textbook Question

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.