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)
4. Applications of Derivatives
Differentials
2:29 minutesProblem 4.7.47
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_v→3 (v-1-√(v²-5)) / (v-3)
Verified step by step guidance1
First, substitute v = 3 into the limit expression to check if it results in an indeterminate form. You will find that both the numerator and denominator become zero, indicating a 0/0 indeterminate form.
Since the limit results in a 0/0 indeterminate form, l'Hôpital's Rule can be applied. This rule states that if the limit of f(v)/g(v) as v approaches a certain value results in 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.
Differentiate the numerator, which is (v - 1 - √(v² - 5)). The derivative of v - 1 is 1, and the derivative of -√(v² - 5) is -1/(2√(v² - 5)) * (2v) using the chain rule.
Differentiate the denominator, which is simply v - 3. The derivative of v - 3 is 1.
Apply l'Hôpital's Rule by taking the limit of the new expression formed by the derivatives: lim_v→3 [1 - v/√(v² - 5)] / 1. Substitute v = 3 into this expression to evaluate the limit.
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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 a quotient of two functions yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, then re-evaluating the limit. This technique simplifies the process of finding limits in complex expressions.
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Square Root Functions
Square root functions, such as √(v²-5), are important in calculus as they can introduce complexities in limit evaluations. Understanding how to manipulate and simplify expressions involving square roots is crucial, especially when approaching limits that may lead to indeterminate forms. Recognizing the behavior of square root functions near specific values helps in applying techniques like L'Hôpital's Rule effectively.
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