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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.67
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|

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1
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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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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Related Practice
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Use the precise definition of infinite limits to prove the following limits.


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