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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 2.4.7a
Chapter 2, Problem 2.4.7a

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>
lim x→1^− f(x)

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1
Identify that the limit \( \lim_{x \to 1^-} f(x) \) involves approaching the point \( x = 1 \) from the left side.
Recognize that a vertical asymptote at \( x = 1 \) implies that as \( x \) approaches 1, \( f(x) \) tends to either positive or negative infinity.
Since we are approaching from the left (\( x \to 1^- \)), examine the behavior of \( f(x) \) as \( x \) gets closer to 1 from values less than 1.
Consider the graph's trend as \( x \to 1^- \): if \( f(x) \) increases without bound, the limit is positive infinity; if it decreases without bound, the limit is negative infinity.
Conclude the analysis by determining the direction of \( f(x) \) as \( x \to 1^- \) based on the graph's behavior.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertical Asymptotes

Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a certain value. In this case, the function f has vertical asymptotes at x=1 and x=2, indicating that as x approaches these values, f(x) does not settle at a finite value but instead diverges.
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Limits

A limit describes the behavior of a function as the input approaches a particular point. The notation lim x→1^− f(x) specifically refers to the limit of f(x) as x approaches 1 from the left side. Understanding limits is crucial for analyzing the behavior of functions near points of discontinuity, such as vertical asymptotes.
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One-Sided Limits

One-Sided Limits

One-sided limits evaluate the behavior of a function as the input approaches a specific point from one direction only. The notation lim x→1^− f(x) indicates that we are interested in the limit as x approaches 1 from values less than 1. This concept is essential for understanding how functions behave near points of discontinuity, particularly when vertical asymptotes are present.
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Related Practice
Textbook Question

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).


a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

Textbook Question

Complete the following sentences in terms of a limit.


a. A function is continuous from the left at a if _____.

Textbook Question

Determine the following limits.


a. limx→1+x−3x2−5x+4{\(\displaystyle\]\lim\)_{x\(\to\)1^{+}}\(\frac{x-3}{\sqrt{x^2-5x+4}\)}}

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


a. The graph of a function can never cross one of its horizontal asymptotes.

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Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x2 − 9)/(x(x−3))

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.


a. lim x→−2^+ f(x)