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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 2.87
Chapter 2, Problem 2.87

Use an appropriate limit definition to prove the following limits.


lim x→ 5x^2 − 25 / x − 5=10

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1
Identify the limit expression: \( \lim_{{x \to 5}} \frac{{x^2 - 25}}{{x - 5}} \).
Recognize that the expression \( x^2 - 25 \) can be factored as \((x - 5)(x + 5)\).
Rewrite the limit expression using the factorization: \( \lim_{{x \to 5}} \frac{{(x - 5)(x + 5)}}{{x - 5}} \).
Cancel the common factor \((x - 5)\) in the numerator and denominator, simplifying the expression to \( \lim_{{x \to 5}} (x + 5) \).
Evaluate the limit by direct substitution: substitute \(x = 5\) into the simplified expression \(x + 5\).

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Key Concepts

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

Limit Definition

The limit definition in calculus refers to the formal approach to finding the limit of a function as it approaches a certain point. It involves evaluating the behavior of the function as the input values get arbitrarily close to a specified value, often denoted as 'a'. This concept is foundational for understanding continuity, derivatives, and integrals.
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Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial expression into simpler components, or factors, that can be multiplied together to yield the original polynomial. In the context of limits, factoring can help simplify expressions that yield indeterminate forms, such as 0/0, allowing for easier evaluation of the limit.
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Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions that do not provide clear information about the limit's value, such as 0/0 or ∞/∞. Recognizing these forms is crucial, as they often require additional techniques, like factoring or L'Hôpital's Rule, to resolve and find the actual limit.
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Related Practice
Textbook Question

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.

f(−1)=−2f\(\left\)(-1\(\right\))=-2, f(1)=2f\(\left\)(1\(\right\))=2, f(0)=0f\(\left\)(0\(\right\))=0, limx→∞f(x)=1{\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)=1}}, limx→−∞f(x)=−1{\(\displaystyle\)\(\lim\)_{x\(\to\)-\(\infty\)}{f(x)=-1}}

Textbook Question

Consider the position function s(t) =−16t^2+100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.

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Textbook Question

b. Estimate a solution to the equation in the given interval using a root finder.


x^5+7x+5=0; (−1,0)

Textbook Question

Use the precise definition of infinite limits to prove the following limits.


limx→41(x−4)2=∞{\(\displaystyle\)\(\lim\)_{x\(\to\)4}}\(\frac{1}{\left(x-4\right)^2}\)=\(\infty\)

Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=x^2−3x+2 / x^10−x^9

Textbook Question

Estimate the following limits using graphs or tables.

limh→0ln(1+h)h{\(\displaystyle\)\(\lim\)_{h\(\to\)0}}\(\frac{\ln\left(1+h\right)}{h}\)

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