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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 64
Chapter 4, Problem 64

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (x - √(x²+4x))

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1
Identify the form of the limit as x approaches infinity. The expression x - √(x²+4x) is of the indeterminate form ∞ - ∞.
To resolve the indeterminate form, multiply and divide the expression by the conjugate: (x - √(x²+4x)) * (x + √(x²+4x)) / (x + √(x²+4x)).
Simplify the numerator using the difference of squares: (x² - (x²+4x)) = -4x.
The expression now becomes -4x / (x + √(x²+4x)). Simplify this expression by dividing the numerator and the denominator by x.
Evaluate the limit of the simplified expression as x approaches infinity. The limit of -4 / (1 + √(1 + 4/x)) as x approaches infinity can be found by considering the behavior of each term.

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Key Concepts

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

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior at points where it may not be explicitly defined, such as infinity or points of discontinuity. Evaluating limits is crucial for determining the continuity and differentiability of functions.
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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. This process can be repeated if the result remains indeterminate.
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Square Root Functions

Square root functions, such as √(x² + 4x), are important in calculus as they often appear in limit problems and can complicate the evaluation of limits. Understanding how to manipulate and simplify expressions involving square roots is essential for applying techniques like l'Hôpital's Rule effectively. Recognizing the behavior of these functions as x approaches infinity is key to solving limit problems.
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The figure shows six containers, each of which is filled from the top. Assume water is poured into the containers at a constant rate and each container is filled in 10 s. Assume also that the horizontal cross sections of the containers are always circles. Let h (t) be the depth of water in the container at time t, for 0 ≤ t ≤ 10 . <IMAGE>


d. For each container, where does h' (the derivative of h ) have an absolute maximum on [0 , 10]?