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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)

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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.

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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.
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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.
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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.
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Related Practice
Textbook Question

Snell’s Law Suppose a light source at A is in a medium in which light travels at a speed v₁ and that point B is in a medium in which light travels at a speed v₂ (see figure). Using Fermat’s Principle, which states that light travels along the path that requires the minimum travel time (Exercise 55), show that the path taken between points A and B satisfies (sinΘ₁/v₁ = (sin Θ₂) /v₂ . <IMAGE>

Textbook Question

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Textbook Question

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Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


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Textbook Question

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