Textbook Question
Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).
lim x → -7 f(x)
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 46d
Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).
lim x→0 f(x)
Verified step by step guidance1
Identify the vertical asymptotes by finding the values of x that make the denominator zero.
Factor the denominator: x^4 - 49x^2 can be rewritten as x^2(x^2 - 49).
Further factor x^2 - 49 as (x - 7)(x + 7) using the difference of squares.
The denominator is now x^2(x - 7)(x + 7). Set each factor equal to zero to find the vertical asymptotes: x^2 = 0, x - 7 = 0, x + 7 = 0.
The vertical asymptotes are at x = 0, x = 7, and x = -7.
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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 in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, evaluating the limit as x approaches 0 helps determine the behavior of the function f(x) near that point, which is crucial for understanding continuity and potential asymptotic behavior.
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Vertical Asymptotes
Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value. For the function f(x) = (x + 7) / (x^4 - 49x^2), vertical asymptotes can be found by identifying values of x that make the denominator zero, leading to undefined behavior in the function.
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Factoring Polynomials
Factoring polynomials is a technique used to simplify expressions and find roots. In the case of the denominator x^4 - 49x^2, factoring can reveal the critical points where the function may have vertical asymptotes. Recognizing that this expression can be factored as x^2(x^2 - 49) aids in identifying the values of x that lead to undefined behavior.
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Related Practice
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