Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.7.20
Chapter 4, Problem 4.7.20

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0 (eˣ - 1) / (2x + 5)

Verified step by step guidance
1
First, substitute x = 0 into the limit expression to check if it results in an indeterminate form. The expression becomes (e^0 - 1) / (2*0 + 5), which simplifies to 0/5 = 0. Since this is not an indeterminate form, l'Hôpital's Rule is not necessary.
Since the limit does not result in an indeterminate form, evaluate the expression directly by substituting x = 0. The expression becomes (e^0 - 1) / (2*0 + 5).
Simplify the expression: e^0 is 1, so the numerator becomes 1 - 1 = 0. The denominator is 2*0 + 5 = 5.
The limit simplifies to 0/5, which is 0.
Thus, the limit of (eˣ - 1) / (2x + 5) as x approaches 0 is 0.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is essential for determining continuity, derivatives, and integrals.
Recommended video:
05:50
One-Sided Limits

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit. This technique simplifies the process of finding limits in complex scenarios.
Recommended video:
Guided course
5:50
Power Rules

Exponential Functions

Exponential functions, such as eˣ, are functions where a constant base is raised to a variable exponent. They are crucial in calculus due to their unique properties, including their continuous growth and the fact that the derivative of eˣ is eˣ itself. Understanding the behavior of exponential functions near specific points, like x = 0, is vital for evaluating limits involving these functions.
Recommended video:
6:13
Exponential Functions
Related Practice
Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


p(t) = 2t³ + 3t² - 36t

183
views
Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (csc² Θ + 2Θ² - 3Θ) dΘ

127
views
Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = cos 2x - x² + 2x

228
views
Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = cos² x on [-π,π]

227
views
Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx

79
views
Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→∞ (e¹/ₓ - 1)/(1/x)

201
views