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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 2.5.63a
Chapter 2, Problem 2.5.63a

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.

Verified step by step guidance
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Step 1: Understand the concept of horizontal asymptotes. A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as the input (x) either goes to positive or negative infinity.
Step 2: Consider the behavior of rational functions. For example, the function f(x) = \(\frac{x}{x^2 + 1}\) has a horizontal asymptote at y = 0.
Step 3: Analyze the graph of f(x) = \(\frac{x}{x^2 + 1}\). As x approaches infinity or negative infinity, the function approaches the horizontal asymptote y = 0. However, for finite values of x, the function can cross the horizontal asymptote.
Step 4: Provide a counterexample. For instance, the function f(x) = \(\frac{x}{x^2 + 1}\) crosses the horizontal asymptote y = 0 at x = 0, since f(0) = 0.
Step 5: Conclude that the statement is false. A function can indeed cross its horizontal asymptote, as demonstrated by the counterexample.

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Key Concepts

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

Horizontal Asymptotes

A horizontal asymptote is a line that a graph approaches as the input values (x) approach positive or negative infinity. It indicates the behavior of a function at extreme values. For example, the function f(x) = 1/x has a horizontal asymptote at y = 0, meaning as x becomes very large or very small, the function values get closer to 0.
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Behavior of Functions Near Asymptotes

Functions can behave differently near their asymptotes. While a function may approach a horizontal asymptote, it is not restricted from crossing it at finite values of x. For instance, the function f(x) = sin(x)/x approaches the horizontal asymptote y = 0 as x approaches infinity, but it crosses the x-axis multiple times.
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True/False Statements in Mathematics

In mathematics, determining the truth of a statement often requires understanding definitions and properties. A statement can be true in some contexts and false in others. In this case, the assertion that a function cannot cross its horizontal asymptote is false, as demonstrated by functions that do cross their asymptotes at finite points.
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Related Practice
Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).

f(x) = (x^4−1)/(x^2−1)

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Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

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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⌋.

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Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


a. Graph the position function, for 0≤t≤9.

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Textbook Question

Let g(x)=x3−4x8∣x−2∣g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\). <IMAGE>

Calculate g(x)g\(\left\)(x\(\right\)) for each value of xx in the following table.

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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.


b. lim x→−2 f(x)

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