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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.R.35
Chapter 2, Problem 2.R.35

Determine the following limits.
lim x→∞ (2x − 3) / (4x + 10)

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1
Identify the highest degree terms in the numerator and the denominator. In this case, both are linear terms: \$2x\( in the numerator and \)4x$ in the denominator.
Divide every term in the numerator and the denominator by \(x\), the highest power of \(x\) present in the expression.
Rewrite the expression as \(\frac{2x/x - 3/x}{4x/x + 10/x}\), which simplifies to \(\frac{2 - 3/x}{4 + 10/x}\).
As \(x\) approaches infinity, the terms \$3/x\( and \)10/x$ approach zero.
Evaluate the limit of the simplified expression \(\frac{2 - 0}{4 + 0}\), which simplifies to \(\frac{2}{4}\).

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Key Concepts

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

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave for very large values of x, which can help determine horizontal asymptotes and overall end behavior.
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One-Sided Limits

Rational Functions

A rational function is a ratio of two polynomials. In the limit problem presented, recognizing that both the numerator and denominator are polynomials allows us to simplify the expression by focusing on the leading terms, which dominate the behavior as x approaches infinity.
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Intro to Rational Functions

Leading Coefficients

The leading coefficients of the highest degree terms in the numerator and denominator play a key role in determining the limit of a rational function as x approaches infinity. For the limit in question, the leading terms (2x and 4x) dictate the limit's value, allowing for straightforward simplification.
Recommended video:
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Introduction to Polynomial Functions
Related Practice
Textbook Question

The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground. Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].

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Textbook Question

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


x=cos x; (0,π/2)

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Textbook Question

Suppose the rental cost for a snowboard is \$25 for the first day (or any part of the first day) plus \$15 for each additional day (or any part of a day).

e. For what values of t is f continuous? Explain.

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Textbook Question

Evaluate limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}.


f(x)=1−e−2xf\(\left\)(x\(\right\))=1-e^{-2x}

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Textbook Question

Determine the following limits.

lim x→∞ (3 tan-1 x + 2)

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Textbook Question

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

g(x)=cos(ex)g\(\left\)(x\(\right\))=\(\cos\]\left\)(e^{x}\(\right\))

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