Textbook Question
Complete the following steps for each function.
c. State the interval(s) of continuity.
f(x)={x^3+4x+1 if x≤0
2x^3 if x>0; a=0
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 2, Problem 41
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→9 √x − 3 / x − 9
Verified step by step guidance1
Recognize that the limit \( \lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} \) is an indeterminate form \( \frac{0}{0} \).
To resolve the indeterminate form, use algebraic manipulation. Multiply the numerator and the denominator by the conjugate of the numerator: \( \sqrt{x} + 3 \).
This gives: \( \lim_{x \to 9} \frac{(\sqrt{x} - 3)(\sqrt{x} + 3)}{(x - 9)(\sqrt{x} + 3)} \).
Simplify the expression: the numerator becomes \( x - 9 \) because \((\sqrt{x} - 3)(\sqrt{x} + 3) = x - 9\).
Cancel \( x - 9 \) from the numerator and denominator, resulting in \( \lim_{x \to 9} \frac{1}{\sqrt{x} + 3} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of a function as x approaches 9.
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Indeterminate Forms
Indeterminate forms occur in calculus when direct substitution in a limit leads to an ambiguous result, such as 0/0. In the given limit, substituting x = 9 results in this form, necessitating further analysis, such as algebraic manipulation or applying L'Hôpital's Rule to resolve the limit.
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Rationalizing the Numerator
Rationalizing the numerator is a technique used to simplify expressions involving square roots. By multiplying the numerator and denominator by the conjugate of the numerator, we can eliminate the square root and simplify the limit calculation. This method is particularly useful when dealing with limits that yield indeterminate forms.
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Related Practice
Textbook Question
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→−1 (2x − 1)^2 − 9 / x + 1
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Textbook Question
Evaluate each limit and justify your answer.
lim x→0 (x / √16x+1-1)^1/3
Textbook Question
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim t→5 (1/t^2 − 4t − 5 −1/ 6(t − 5))
Textbook Question
Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?
f(x)=√x^2−3x+2
Textbook Question
Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?
f(x)=√25−x^2
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