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 2.2.38
Chapter 2, Problem 2.2.38

Limits of quotients


Find the limits in Exercises 23–42.


limx→−1 (√(x² + 8) − 3) / (x + 1)

Verified step by step guidance
1
Recognize that the limit involves a quotient where direct substitution results in an indeterminate form 0/0. This suggests the need for algebraic manipulation.
Multiply the numerator and the denominator by the conjugate of the numerator, which is (√(x² + 8) + 3), to eliminate the square root in the numerator.
Simplify the expression by applying the difference of squares formula: (a - b)(a + b) = a² - b². Here, a = √(x² + 8) and b = 3.
After simplification, the numerator becomes (x² + 8) - 9, which simplifies further to x² - 1. Notice that x² - 1 can be factored as (x - 1)(x + 1).
Cancel the common factor (x + 1) from the numerator and the denominator, then evaluate the limit of the simplified expression as x approaches -1.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

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 limit of a quotient as x approaches -1. Understanding limits helps in analyzing the behavior of functions near points of interest, especially where they may not be directly computable.
Recommended video:
05:50
One-Sided Limits

Rational Functions

A rational function is a ratio of two polynomials. In the given limit problem, the expression involves a square root in the numerator and a linear polynomial in the denominator. Recognizing the structure of rational functions is crucial for applying limit laws and techniques, such as factoring or rationalizing, to simplify the expression before evaluating the limit.
Recommended video:
6:04
Intro to Rational Functions

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. If direct substitution in the limit leads to such forms, L'Hôpital's Rule allows us to differentiate the numerator and denominator separately. This technique is particularly useful in the given problem, where direct substitution results in an indeterminate form, necessitating further analysis.
Recommended video:
Guided course
5:50
Power Rules
Related Practice
Textbook Question

Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = 1/x, x = -2

Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


lim x→1 1/x = 1

Textbook Question

At what points are the functions in Exercises 13–30 continuous?


g(x) = { (x² − x – 6)/(x – 3), x ≠ 3

5, x = 3


Textbook Question

At what points are the functions in Exercises 13–30 continuous?

y = 1/(x – 2) – 3x

Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x → ∞ √((8x² − 3) / (2x² + x))

Textbook Question

Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.