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 45c
Analyze the following limits and find the vertical asymptotes of f(x) =(x − 5) / (x2 − 25).
lim x→−5+ f(x)
Verified step by step guidance1
Step 1: Identify the points where the function f(x) = \( \frac{x - 5}{x^2 - 25} \) is undefined. This occurs when the denominator is zero. Set the denominator equal to zero: \( x^2 - 25 = 0 \).
Step 2: Solve the equation \( x^2 - 25 = 0 \) to find the values of x that make the denominator zero. This can be factored as \( (x - 5)(x + 5) = 0 \), giving the solutions \( x = 5 \) and \( x = -5 \).
Step 3: Determine if these points are vertical asymptotes by checking the behavior of the function as x approaches these values. Since the numerator \( x - 5 \) is zero at \( x = 5 \), the function has a removable discontinuity at \( x = 5 \), not a vertical asymptote.
Step 4: Analyze the limit \( \lim_{x \to -5^+} f(x) \). As x approaches -5 from the right, the denominator \( x^2 - 25 \) approaches zero, and the numerator \( x - 5 \) approaches -10. This indicates a vertical asymptote at \( x = -5 \).
Step 5: Conclude that the function has a vertical asymptote at \( x = -5 \) because the limit of f(x) as x approaches -5 from the right is either positive or negative infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits describe the behavior of a function as the input approaches a certain value. In this context, evaluating the limit as x approaches -5 from the right (denoted as x→−5⁺) helps determine how the function behaves near that point, which is crucial for identifying vertical asymptotes.
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One-Sided Limits
Vertical Asymptotes
Vertical asymptotes occur at values of x where a function approaches infinity or negative infinity, typically where the denominator of a rational function equals zero while the numerator does not. For the function f(x) = (x − 5) / (x² − 25), finding the points where the denominator is zero will help identify potential vertical asymptotes.
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Factoring Polynomials
Factoring polynomials is a method used to simplify expressions and find roots. In this case, the denominator x² − 25 can be factored into (x − 5)(x + 5), which reveals the points where the function may have vertical asymptotes, specifically at x = 5 and x = -5.
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Related Practice
Textbook Question
Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).
lim x→0 f(x)
Textbook Question
Analyze the following limits and find the vertical asymptotes of f(x) = (x − 5) / (x2 − 25).
lim x → -5- f(x)
Textbook Question
Sketch the graph of a function with the given properties. You do not need to find a formula for the function.
f(2) = 1,lim x→2 f(x) = 3
Textbook Question
Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?
f(x)=√x^2−3x+2
Textbook Question
Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?
f(t)=(t^2−1)^3/2
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