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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.R57
Chapter 2, Problem 2.R57

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}

Verified step by step guidance
1
Identify the function given: \( f(x) = 1 - e^{-2x} \). We need to evaluate the limits as \( x \to \infty \) and \( x \to -\infty \).
Consider the limit \( \lim_{x \to \infty} f(x) \). As \( x \to \infty \), the term \( e^{-2x} \) approaches 0 because the exponent \(-2x\) becomes very large and negative, making \( e^{-2x} \) very small.
Thus, \( \lim_{x \to \infty} f(x) = 1 - 0 = 1 \).
Now, consider the limit \( \lim_{x \to -\infty} f(x) \). As \( x \to -\infty \), the term \( e^{-2x} \) approaches infinity because the exponent \(-2x\) becomes very large and positive.
Therefore, \( \lim_{x \to -\infty} f(x) = 1 - \infty = -\infty \).

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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. Evaluating limits at infinity often involves simplifying the function to identify dominant terms that dictate its behavior.
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Exponential Functions

Exponential functions, such as f(x) = 1 - e^(-2x), are characterized by a constant base raised to a variable exponent. These functions exhibit rapid growth or decay, depending on the sign of the exponent. Understanding their properties, including their limits as x approaches infinity or negative infinity, is essential for analyzing their long-term behavior and applications in various fields.
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Exponential Functions

Continuous Functions

A function is continuous if there are no breaks, jumps, or holes in its graph. This property is important when evaluating limits, as continuous functions allow for the direct substitution of values. In the context of limits at infinity, continuity ensures that the limit can be determined by examining the function's behavior without encountering undefined points.
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Intro to Continuity
Related Practice
Textbook Question

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).


a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

Textbook Question

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


x=cos x; (0,π/2)

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.

Textbook Question

Determine the following limits.


a. limx→1+x−3x2−5x+4{\(\displaystyle\]\lim\)_{x\(\to\)1^{+}}\(\frac{x-3}{\sqrt{x^2-5x+4}\)}}

Textbook Question

Let g(x)={5x−2if x<1aif x=1ax2+bxif x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)5x-2 & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ ax^2+bx & \(\text{if }\)x>1\(\end{cases}\).


Determine values of the constants aa and bb , if possible, for which gg is continuous at x=1x=1.

Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x2 − 9)/(x(x−3))