Textbook Question
Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.
d. lim x→0^+ f(x)
1. Limits and Continuity
Finding Limits Algebraically
2:14 minutesProblem 21a
Textbook Question
Determine the following limits.
a. lim x→2^+ 1 x − 2
Verified step by step guidance1
Identify the type of limit: This is a one-sided limit as \( x \) approaches 2 from the right (denoted by \( x \to 2^+ \)).
Recognize the form of the function: The function is \( \frac{1}{x - 2} \).
Consider the behavior of \( x - 2 \) as \( x \to 2^+ \): As \( x \) approaches 2 from the right, \( x - 2 \) becomes a small positive number.
Analyze the behavior of the function: Since \( x - 2 \) is a small positive number, \( \frac{1}{x - 2} \) becomes a large positive number.
Conclude the limit: The limit of \( \frac{1}{x - 2} \) as \( x \to 2^+ \) is positive infinity.
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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 or infinity. The notation 'lim x→a f(x)' indicates the value that f(x) approaches as x gets arbitrarily close to a.
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One-Sided Limits
One-Sided Limits
One-sided limits are used to evaluate the behavior of a function as the input approaches a specific value from one side only. The notation 'lim x→a^+' refers to the limit as x approaches a from the right (values greater than a), while 'lim x→a^-' refers to the limit as x approaches a from the left (values less than a). This distinction is crucial for understanding functions that may behave differently on either side of a point.
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Infinite Limits
An infinite limit occurs when the value of a function increases or decreases without bound as the input approaches a certain point. In the context of the given limit, 'lim x→2^+ 1/(x - 2)', as x approaches 2 from the right, the denominator approaches zero, causing the function to approach positive infinity. This concept is essential for analyzing vertical asymptotes in rational functions.
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