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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 2.59
Chapter 2, Problem 2.59

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


f(x)=6ex+203ex+4f\(\left\)(x\(\right\))=\(\frac{6e^{x}\)+20}{3e^{x}+4}

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Identify the function for which you need to find the limits: \( f(x) = \frac{6e^{x} + 20}{3e^{x} + 4} \).
To evaluate \( \lim_{x \to \infty} f(x) \), divide the numerator and the denominator by \( e^x \), the highest power of \( e \) in the expression. This simplifies the function to \( \frac{6 + \frac{20}{e^x}}{3 + \frac{4}{e^x}} \).
As \( x \to \infty \), the terms \( \frac{20}{e^x} \) and \( \frac{4}{e^x} \) approach 0 because \( e^x \) grows exponentially. Thus, the expression simplifies to \( \frac{6}{3} \).
Now, evaluate \( \lim_{x \to -\infty} f(x) \). In this case, \( e^x \to 0 \) as \( x \to -\infty \), so the function simplifies to \( \frac{20}{4} \).
Conclude by stating the limits: \( \lim_{x \to \infty} f(x) = 2 \) and \( \lim_{x \to -\infty} f(x) = 5 \).

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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 describe the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine horizontal asymptotes and the end behavior of functions.
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Cases Where Limits Do Not Exist

Exponential Functions

Exponential functions, such as f(x) = (6e^x + 20)/(3e^x + 4), are characterized by a constant base raised to a variable exponent. These functions grow rapidly as x increases, making them essential in calculus for evaluating limits and understanding growth rates.
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Exponential Functions

Rational Functions

Rational functions are ratios of polynomials, which can exhibit unique behaviors at different values of x. In the context of limits, analyzing the degrees of the numerator and denominator helps determine the limit as x approaches infinity or negative infinity, revealing important characteristics of the function.
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Intro to Rational Functions
Related Practice
Textbook Question

Determine the following limits.


limθ→0−sinθcos2θ−1{\(\displaystyle\[\lim\)_{\(\theta\]\to\)0^{-}}\(\frac{\sin\theta}{\cos^2\theta-1}\)}

Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2+x−2 / x−1; a=1

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Textbook Question

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


limx→0(1x2+1)=∞{\(\displaystyle\]\lim\)_{x\(\to\)0}}\(\left\)(\(\frac{1}{x^2}\)+1\(\right\))=\(\infty\)

Textbook Question

Determine the following limits.


limθ→02+sinθ1−cos2θ{\(\displaystyle\[\lim\)_{\(\theta\]\to\)0}}\(\frac{2+\sin\theta}{1-\cos^2\theta}\)

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Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

f(x)=−3e−xf\(\left\)(x\(\right\))=-3e^{-x}

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Textbook Question

Explain the meaning of lim x→a f(x) =∞.