Skip to main content
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 81a
Chapter 2, Problem 81a

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


f(x)=cos x+2√x / √x.

Verified step by step guidance
1
Step 1: Simplify the function f(x) = \(\frac{\cos x + 2\sqrt{x}\)}{\(\sqrt{x}\)}.
Step 2: Divide each term in the numerator by \(\sqrt{x}\) to get f(x) = \(\frac{\cos x}{\sqrt{x}\)} + 2.
Step 3: Analyze \(\lim\)_{x \(\to\) \(\infty\)} f(x). As x approaches infinity, \(\frac{\cos x}{\sqrt{x}\)} approaches 0 because \(\cos\) x is bounded between -1 and 1, while \(\sqrt{x}\) grows without bound.
Step 4: Therefore, \(\lim\)_{x \(\to\) \(\infty\)} f(x) = 0 + 2 = 2, indicating a horizontal asymptote at y = 2 as x approaches infinity.
Step 5: Analyze \(\lim\)_{x \(\to\) -\(\infty\)} f(x). Since \(\sqrt{x}\) is not defined for negative x, the limit as x approaches negative infinity does not exist, and there is no horizontal asymptote in this direction.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. This analysis helps determine the end behavior of the function, which is crucial for identifying horizontal asymptotes. For example, if the limit of f(x) as x approaches infinity exists and is a finite number, it indicates a horizontal asymptote at that value.
Recommended video:
05:50
One-Sided Limits

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as x approaches infinity or negative infinity. They represent the value that the function approaches but may never actually reach. To find horizontal asymptotes, one typically evaluates the limits of the function at both ends of the x-axis, determining if the function stabilizes at a particular value.
Recommended video:
5:46
Graphs of Exponential Functions

Rational Functions and Simplification

Rational functions are ratios of polynomials, and their behavior at infinity can often be simplified by dividing the numerator and denominator by the highest power of x present. In the given function, simplifying helps to clearly see how the function behaves as x approaches infinity or negative infinity, which is essential for accurately determining limits and asymptotes.
Recommended video:
6:04
Intro to Rational Functions
Related Practice
Textbook Question

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→∞ cot^−1

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)=cos x+2√x / √x.

1
views
Textbook Question

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


f(x)=3e^x+10 / e^x

Textbook Question

A right circular cylinder with a height of 10 cm and a surface area of S cm2 has a radius given by r(S)=1/2(√100+2S/π −10).

Find lim S→0^+ r(S) and interpret your result.

1
views
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)=3e^x+10 / e^x

1
views
Textbook Question

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→−∞ cot^−1x

1
views