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 2.8.84
Chapter 2, Problem 2.8.84

Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → ∞ (√(x + 9) − √(x + 4))

Verified step by step guidance
1
Identify the expression whose limit you need to find: \( \lim_{x \to \infty} (\sqrt{x + 9} - \sqrt{x + 4}) \).
To simplify the expression, multiply and divide by the conjugate: \( \frac{(\sqrt{x + 9} - \sqrt{x + 4})(\sqrt{x + 9} + \sqrt{x + 4})}{\sqrt{x + 9} + \sqrt{x + 4}} \).
The numerator becomes a difference of squares: \((x + 9) - (x + 4) = 5\).
The expression simplifies to \( \frac{5}{\sqrt{x + 9} + \sqrt{x + 4}} \).
As \( x \to \infty \), both \( \sqrt{x + 9} \) and \( \sqrt{x + 4} \) approach \( \sqrt{x} \), so the denominator approaches \( 2\sqrt{x} \). Thus, the limit becomes \( \lim_{x \to \infty} \frac{5}{2\sqrt{x}} \), which approaches 0.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m

Key Concepts

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the behavior of the function as x approaches infinity, which helps us understand the end behavior of the function and how it simplifies under such conditions.
Recommended video:
05:50
One-Sided Limits

Conjugates

The conjugate of a binomial expression is formed by changing the sign between two terms. In the context of limits, multiplying by the conjugate can help eliminate square roots or simplify expressions, making it easier to evaluate the limit. This technique is particularly useful when dealing with differences of square roots.
Recommended video:
06:13
Limits of Rational Functions with Radicals

Rationalization

Rationalization is a method used to eliminate radicals from the denominator or simplify expressions involving square roots. By multiplying the numerator and denominator by the conjugate, we can transform the expression into a more manageable form, allowing for easier calculation of limits, especially as x approaches infinity.
Recommended video:
6:04
Intro to Rational Functions
Related Practice
Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


lim x→0 x sin (1/x) = 0


<IMAGE>

Textbook Question

Domains and Asymptotes


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


y = 4 + 3x² / (x² + 1)

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limh→0− h / sin 3h

1
views
Textbook Question

Horizontal and Vertical Asymptotes


Determine the domain and range of y = (√16―x²) / (x―2).

1
views
Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→1 f(x) = 1 if f(x) = {x², x ≠ 1

2, x = 1

Textbook Question

Limits with trigonometric functions


Find the limits in Exercises 43–50.


limx→−π √(x + 4) cos(x + π)