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 39
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
Verified step by step guidance1
Identify the limit expression: \( \lim_{{x \to -1}} \frac{{(2x - 1)^2 - 9}}{{x + 1}} \).
Notice that direct substitution of \( x = -1 \) results in an indeterminate form \( \frac{0}{0} \).
Factor the numerator: \((2x - 1)^2 - 9\) is a difference of squares, which can be factored as \((2x - 1 - 3)(2x - 1 + 3)\).
Simplify the expression: \((2x - 1 - 3)(2x - 1 + 3) = (2x - 4)(2x + 2)\).
Cancel the common factor \(x + 1\) from the numerator and denominator, then evaluate the limit by substituting \(x = -1\) into the simplified expression.
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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. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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Factoring and Simplifying Expressions
Factoring involves rewriting an expression as a product of its factors, which can simplify the evaluation of limits, especially when direct substitution leads to indeterminate forms like 0/0. In the given limit, factoring the numerator allows for cancellation with the denominator, making it easier to compute the limit as x approaches -1.
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Indeterminate Forms
Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is essential, as they often require additional techniques, such as factoring, rationalizing, or applying L'Hôpital's Rule, to resolve and find the actual limit.
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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→b (x − b)^50 − x + b / x − b
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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→9 √x − 3 / x − 9
Textbook Question
Determine the following limits.
lim x→−∞ (e^x cos x +3)
Textbook Question
Evaluate each limit and justify your answer.
lim x→0 (x / √16x+1-1)^1/3
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
11
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