Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
a. limx→0+ (1 − cos x) / |cos x − 1|
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1. Limits and Continuity
Finding Limits Algebraically
1:56 minutesProblem 55b
Textbook Question
Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim (x²/2 − 1/x) as
b. x→0⁻
Verified step by step guidance1
Step 1: Understand the problem. We need to find the limit of the function \( \frac{x^2}{2} - \frac{1}{x} \) as \( x \) approaches 0 from the left (denoted as \( x \to 0^- \)). This means we are considering values of \( x \) that are slightly less than 0.
Step 2: Analyze the behavior of each term in the function as \( x \to 0^- \). The term \( \frac{x^2}{2} \) approaches 0 because \( x^2 \) becomes very small as \( x \) approaches 0, and dividing by 2 does not affect the limit.
Step 3: Consider the term \( \frac{1}{x} \). As \( x \to 0^- \), \( x \) is negative and very close to 0, so \( \frac{1}{x} \) becomes very large negatively (approaches \( -\infty \)).
Step 4: Combine the behavior of both terms. The term \( \frac{x^2}{2} \) approaches 0, while \( \frac{1}{x} \) approaches \( -\infty \). Therefore, the expression \( \frac{x^2}{2} - \frac{1}{x} \) will be dominated by the \( -\frac{1}{x} \) term, which approaches \( -\infty \).
Step 5: Conclude the limit. Since the dominant term \( -\frac{1}{x} \) approaches \( -\infty \), the limit of the entire expression \( \frac{x^2}{2} - \frac{1}{x} \) as \( x \to 0^- \) is \( -\infty \).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of the function as x approaches 0 from the left (denoted as x→0⁻).
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One-Sided Limits
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only. The notation x→0⁻ indicates that we are considering values of x that are less than 0. This is crucial for determining the limit of functions that may behave differently when approached from the left versus the right.
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Indeterminate Forms
Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In the given limit problem, as x approaches 0, the term 1/x becomes significant, potentially leading to an indeterminate form that requires further analysis to resolve. Understanding how to manipulate these forms is essential for finding limits.
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