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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 2.13
Chapter 2, Problem 2.13

Determine the following limits at infinity.


lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

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Consider the function \(e^t\). As \(t\) approaches infinity, \(e^t\) grows exponentially without bound. Therefore, \(\lim_{t \to \infty} e^t = \infty\).
Now, consider the function \(e^t\) as \(t\) approaches negative infinity. As \(t\) becomes more negative, \(e^t\) approaches zero because the exponential function decays rapidly. Therefore, \(\lim_{t \to -\infty} e^t = 0\).
Next, consider the function \(e^{-t}\) as \(t\) approaches infinity. Notice that \(e^{-t} = \frac{1}{e^t}\). As \(t\) becomes very large, \(e^t\) becomes very large, making \(\frac{1}{e^t}\) approach zero. Therefore, \(\lim_{t \to \infty} e^{-t} = 0\).

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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 refer to the behavior of a function as the input approaches positive or negative infinity. Understanding these limits helps determine the end behavior of functions, which is crucial in calculus for analyzing asymptotic behavior and graphing functions.
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Exponential Functions

Exponential functions, such as e^t, are functions of the form f(t) = a^t, where 'a' is a positive constant. The base 'e' (approximately 2.718) is particularly important in calculus due to its unique properties, including its derivative being equal to the function itself, which simplifies many calculations involving growth and decay.
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Exponential Functions

Behavior of e^t and e^−t

The function e^t grows rapidly as t approaches infinity, leading to a limit of infinity. Conversely, e^−t approaches zero as t approaches infinity, indicating decay. Understanding these behaviors is essential for evaluating the specified limits and interpreting their implications in various contexts.
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Related Practice
Textbook Question

Use the continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.


g(x)=∣x+4x2−4∣g\(\left\)(x\(\right\))=\(\left\[\vert\]\frac{x+4}{x^2-4}\[\right\]\vert\)

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)=sinxf\(\left\)(x\(\right\))=\(\sin\) x

Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→3 −5x / √4x − 3

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Textbook Question

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)

a. Determine the value of a for which gg is continuous from the left at 11.

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→7 f(x)=9, where f(x)={3x−12 if x≤7

x+2 if x>7

Textbook Question

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


limx→0(1x4−sin(x))=∞{\(\displaystyle\[\lim\)_{x\(\to\)0}}\(\left\)(\(\frac{1}{x^4}\)-\(\sin\]\left\)(x\(\right\))\(\right\))=\(\infty\)