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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 108
Chapter 3, Problem 108

Use the definition of the derivative to evaluate the following limits.
limx→25x−25x−2\(\lim\)_{x\(\to\)2}\(\frac{5^{x}\)-25}{x-2}_{}

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Recognize that the limit \( \lim_{x\to2}\frac{5^{x}-25}{x-2} \) is in the indeterminate form \( \frac{0}{0} \), which suggests using L'Hôpital's Rule or algebraic manipulation.
Rewrite the expression \( 5^x - 25 \) as \( 5^x - 5^2 \) to identify a common factor.
Factor the numerator as a difference of powers: \( 5^x - 5^2 = (5-5)(5^{x-1} + 5^{x-2} \cdot 5 + \ldots + 5^2) \).
Apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately: differentiate \( 5^x \) to get \( 5^x \ln(5) \) and \( x-2 \) to get 1.
Evaluate the new limit \( \lim_{x\to2}\frac{5^x \ln(5)}{1} \) by substituting \( x = 2 \) into the expression.

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Key Concepts

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

Definition of the Derivative

The derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is expressed as f'(a) = lim (h→0) [f(a+h) - f(a)] / h. This concept is fundamental for evaluating limits that represent the slope of the tangent line to the curve at a specific point.
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Limit Evaluation

Limits are used to analyze the behavior of functions as they approach a certain point. In this context, we are interested in the limit as x approaches 2 for the expression (5^x - 25) / (x - 2). Evaluating this limit often involves techniques such as direct substitution, factoring, or applying L'Hôpital's Rule when the limit results in an indeterminate form.
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Exponential Functions

Exponential functions, such as 5^x, are functions where a constant base is raised to a variable exponent. They exhibit unique properties, including rapid growth and specific limits as x approaches certain values. Understanding the behavior of exponential functions is crucial for evaluating limits involving expressions like 5^x, especially when determining continuity and differentiability at specific points.
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Related Practice
Textbook Question

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 

a. Determine the average and marginal costs for x = 3000 lawn mowers.

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Textbook Question

Water flows into a conical tank at a rate of 2 ft³/min. If the radius of the top of the tank is 4 ft and the height is 6 ft, determine how quickly the water level is rising when the water is 2 ft deep in the tank.

Textbook Question

Use the definition of the derivative to evaluate the following limits.

limh→0ln(e8+h)−8h\(\lim\)_{h\(\to\)0}\(\frac{\ln\left(e^8+h\right)-8}{h}\)_{}

Textbook Question

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 

b. Interpret the meaning of your results in part (a).

Textbook Question

A spherical balloon is inflated at a rate of 10 cm³/min. At what rate is the diameter of the balloon increasing when the balloon has a diameter of 5 cm?

Textbook Question

Derivatives of even and odd functions Recall that f is even if f(−x) = f(x), for all x in the domain of f, and f is odd if f(−x) = −f(x) for all x in the domain of f.

b. If f is a differentiable, odd function on its domain, determine whether f' is even, odd, or neither.

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