Skip to main content
Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.6.71
Chapter 2, Problem 2.6.71

Domains and Asymptotes


Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.


y = (√(x² + 4)) / x

Verified step by step guidance
1
Step 1: Determine the domain of the function y = (√(x² + 4)) / x. The domain consists of all x-values for which the function is defined. Since the square root function √(x² + 4) is defined for all real numbers, we need to ensure the denominator x is not zero to avoid division by zero. Therefore, the domain is all real numbers except x = 0.
Step 2: Identify vertical asymptotes by examining the behavior of the function as x approaches values that are not in the domain. Since x = 0 is not in the domain, check the limit of y as x approaches 0 from the left and right. If the limit approaches infinity or negative infinity, x = 0 is a vertical asymptote.
Step 3: Determine horizontal asymptotes by evaluating the limit of y as x approaches positive and negative infinity. Calculate lim(x→∞) (√(x² + 4)) / x and lim(x→-∞) (√(x² + 4)) / x. Simplify the expression by dividing the numerator and the denominator by x, which is the highest power of x in the denominator.
Step 4: Simplify the expression (√(x² + 4)) / x by dividing both the numerator and the denominator by x. This results in √(1 + 4/x²). As x approaches infinity, 4/x² approaches 0, so the expression simplifies to √1 = 1. Therefore, the horizontal asymptote is y = 1.
Step 5: Conclude the analysis by summarizing the domain and asymptotes. The domain of the function is all real numbers except x = 0. The function has a vertical asymptote at x = 0 and a horizontal asymptote at y = 1.

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.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function y = (√(x² + 4)) / x, the domain excludes x = 0 because division by zero is undefined. Additionally, the expression under the square root, x² + 4, must be non-negative, which is always true for real numbers.
Recommended video:
5:10
Finding the Domain and Range of a Graph

Limits and Asymptotes

Limits help determine the behavior of a function as the input approaches a particular value, which is crucial for identifying asymptotes. Vertical asymptotes occur where the function becomes undefined, often where the denominator is zero. Horizontal or oblique asymptotes are found by evaluating the limit of the function as x approaches infinity or negative infinity.
Recommended video:
03:07
Cases Where Limits Do Not Exist

Square Root Function

The square root function, √(x), is defined for non-negative values of x. In the context of y = (√(x² + 4)) / x, the expression x² + 4 is always positive, ensuring the square root is defined for all real x. Understanding how the square root affects the function's behavior is essential for analyzing its domain and asymptotic properties.
Recommended video:
03:19
Completing the Square to Rewrite the Integrand Example 6
Related Practice
Textbook Question

Theory and Examples


Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

Textbook Question

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→0 (−1) / (x² (x + 1))

Textbook Question

Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Textbook Question

Graphing Simple Rational Functions


Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.


y = −3/(x − 3)

Textbook Question

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=x³−3x²+4, P(2,0)

Textbook Question

Removable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.