Textbook Question
Evaluate each limit.
lim θ→0 (1/(2+sinθ)-1/2)/sin θ
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Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 2, Problem 57
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→4 3(x − 4)√x + 5 / 3 − √x + 5
Verified step by step guidance1
Identify the limit expression: \( \lim_{{x \to 4}} \frac{3(x - 4)\sqrt{x + 5}}{3 - \sqrt{x + 5}} \).
Substitute \( x = 4 \) into the expression to check if it results in an indeterminate form. If it does, proceed with algebraic manipulation.
Notice that direct substitution gives \( \frac{0}{0} \), an indeterminate form, suggesting the need for further simplification.
Rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator: \( 3 + \sqrt{x + 5} \).
Simplify the expression and evaluate the limit again by substituting \( x = 4 \) after simplification.
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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 how functions behave near specific points, which is crucial for evaluating expressions that may be undefined at those points. In this case, we are interested in the limit as x approaches 4.
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One-Sided Limits
Rational Functions
Rational functions are expressions formed by the ratio of two polynomials. They can exhibit unique behaviors, such as asymptotes or discontinuities, particularly at points where the denominator equals zero. In the given limit problem, recognizing the structure of the function will help in simplifying the expression before evaluating the limit.
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6:04Intro to Rational Functions
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 ∞/∞. It states that if the limit of a function yields such a form, one can take the derivative of the numerator and the derivative of the denominator separately, then re-evaluate the limit. This rule is particularly useful in the context of the given limit problem, where direct substitution may lead to an indeterminate form.
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Guided course
5:50Power Rules
Related Practice
Textbook Question
Evaluate each limit.
lim x→0 e^4x−1 / e^x−1
Textbook Question
Evaluate each limit.
lim x→0+ 1−cos^2x / sin x
Textbook Question
A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.
a. lim x→−2^+ f(x)
Textbook Question
Evaluate each limit.
lim x→e^2 ln^2x−5 ln x+6 lnx−2
Textbook Question
A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.
lim x→−2^− f(x)
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