Textbook Question
Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).
lim x→0 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 46c
Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).
lim x → -7 f(x)
Verified step by step guidance1
Step 1: Identify the points where the denominator is zero, as these are potential vertical asymptotes. Set the denominator equal to zero: \(x^4 - 49x^2 = 0\).
Step 2: Factor the equation \(x^4 - 49x^2 = 0\) by factoring out \(x^2\), resulting in \(x^2(x^2 - 49) = 0\).
Step 3: Further factor \(x^2 - 49\) using the difference of squares: \((x - 7)(x + 7)\). This gives the factored form \(x^2(x - 7)(x + 7) = 0\).
Step 4: Solve for \(x\) in the equation \(x^2(x - 7)(x + 7) = 0\). The solutions are \(x = 0\), \(x = 7\), and \(x = -7\). These are the potential vertical asymptotes.
Step 5: Analyze the limit \(\lim_{x \to -7} \frac{x + 7}{x^4 - 49x^2}\). Substitute \(x = -7\) into the factored form to check if the limit results in an indeterminate form like \(\frac{0}{0}\), which would confirm a vertical asymptote at \(x = -7\).
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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 -7 helps determine the behavior of the function f(x) near that point, which is crucial for identifying vertical asymptotes.
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Vertical Asymptotes
Vertical asymptotes occur in a function when the output approaches infinity or negative infinity as the input approaches a specific value. They are typically found by identifying values of x that make the denominator zero while the numerator remains non-zero, indicating that the function is undefined at those points.
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Factoring Polynomials
Factoring polynomials is a technique used to simplify expressions and identify roots or critical points. In the given function, factoring the denominator x^4 - 49x^2 can reveal the values of x that lead to vertical asymptotes, as it allows for easier identification of where the function is undefined.
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Related Practice
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