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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 2.4.9a
Chapter 2, Problem 2.4.9a

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>
lim x→−2^− h(x)

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1
Identify that the limit is approaching from the left side of x = -2, denoted by x → -2⁻.
Recognize that a vertical asymptote at x = -2 implies that as x approaches -2, the function h(x) tends to infinity or negative infinity.
Since the limit is from the left, consider the behavior of h(x) as x approaches -2 from values less than -2.
Examine the graph of h(x) to determine whether h(x) increases towards positive infinity or decreases towards negative infinity as x approaches -2 from the left.
Conclude the limit based on the observed behavior of h(x) as x approaches -2 from the left side.

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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 h has vertical asymptotes at x = -2 and x = 3, indicating that as x approaches these values, h(x) will diverge to infinity or negative infinity.
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Limits

A limit describes the behavior of a function as the input approaches a particular value. The notation lim x→−2^− h(x) specifically refers to the limit of h(x) as x approaches -2 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→−2^− h(x) indicates a left-hand limit, meaning we are interested in the values of h(x) as x approaches -2 from values less than -2. This concept is essential for understanding how functions behave near points of discontinuity.
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Related Practice
Textbook Question

Complete the following sentences in terms of a limit.


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

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Textbook Question

Determine the following limits.


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

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Textbook Question

Determine the following limits.


a. limx→2+1x(x−2){\(\displaystyle\]\lim\)_{x\(\to\)2^{+}}}\(\frac{1}{\sqrt{x\left(x-2\right)}\)}

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Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

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Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

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Textbook Question

The hyperbolic cosine function, denoted cosh(x)\(\cosh\]\left\)(x\(\right\)), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh(x)=ex+e−x2\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}.


a. Determine its end behavior by analyzing limx→∞cosh(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{\(\cosh\)(x)}} and limx→−∞cosh(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{\(\cosh\)(x)}}.

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