Textbook Question
Which one of the following intervals is not symmetric about x=5?
a.(1, 9)
b.(4, 6)
c.(3, 8)
d.(4.5, 5.5)
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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 2.67
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→3 x − 3 /|x − 3|
Verified step by step guidance1
Identify the expression: \( \lim_{{x \to 3}} \frac{{x - 3}}{{|x - 3|}} \).
Recognize that the expression involves an absolute value, which affects the limit depending on the direction of approach.
Consider the limit as \( x \to 3^+ \) (from the right): In this case, \( x - 3 > 0 \), so \(|x - 3| = x - 3\).
Evaluate the expression \( \frac{{x - 3}}{{x - 3}} \) as \( x \to 3^+ \), which simplifies to 1.
Consider the limit as \( x \to 3^- \) (from the left): In this case, \( x - 3 < 0 \), so \(|x - 3| = -(x - 3)\).
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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 points of interest, including points of discontinuity. In this case, we are examining the limit of a function as x approaches 3, which is crucial for determining the function's value or behavior at that point.
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One-Sided Limits
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is essential in the given limit problem because it affects the behavior of the expression as x approaches 3 from different directions. Understanding how the absolute value function behaves helps in analyzing the limit's outcome, particularly in cases where the function may change its form based on the input.
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One-Sided Limits
One-sided limits refer to the limits of a function as the input approaches a specific value from one side, either the left (denoted as x → c-) or the right (denoted as x → c+). In this problem, evaluating the limit as x approaches 3 from both sides is necessary to determine if the overall limit exists. If the left-hand limit and right-hand limit yield different results, the limit at that point does not exist.
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05:50One-Sided Limits
Related Practice
Textbook Question
Determine the following limits at infinity.
lim x→∞ (5 + 1/x +10/x^2)
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Textbook Question
Evaluate each limit.
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Textbook Question
Evaluate each limit and justify your answer.
lim x→2 (3 / 2x^5−4x^2−50)^4
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Textbook Question
Determine the following limits.
lim x→∞ (x4 − 1) / (x5 + 2)
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Textbook Question
Determine the following limits.