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 43
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))
Verified step by step guidance1
Identify the limit expression: \( \lim_{{t \to 5}} \left( \frac{1}{t^2 - 4t - 5} - \frac{1}{6(t - 5)} \right) \).
Factor the quadratic expression in the denominator: \( t^2 - 4t - 5 = (t - 5)(t + 1) \).
Rewrite the limit expression using the factored form: \( \lim_{{t \to 5}} \left( \frac{1}{(t - 5)(t + 1)} - \frac{1}{6(t - 5)} \right) \).
Combine the fractions over a common denominator: \( \frac{1}{(t - 5)(t + 1)} - \frac{1}{6(t - 5)} = \frac{6 - (t + 1)}{6(t - 5)(t + 1)} \).
Simplify the numerator: \( 6 - (t + 1) = 5 - t \), and then evaluate the limit as \( t \to 5 \).
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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 question, we are tasked with finding the limit of a function as t approaches 5, which requires evaluating the function's behavior close to that point.
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Indeterminate Forms
Indeterminate forms occur in calculus when direct substitution into a limit results in expressions like 0/0 or ∞/∞. These forms require further analysis, often using algebraic manipulation or L'Hôpital's Rule, to resolve the limit. In the given question, substituting t = 5 directly into the expression leads to an indeterminate form, necessitating additional steps to find the limit.
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L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms by differentiating the numerator and denominator. If a limit results in 0/0 or ∞/∞, applying this rule can simplify the expression and help find the limit. In this case, if the limit leads to an indeterminate form, L'Hôpital's Rule may be a suitable approach to determine the limit as t approaches 5.
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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→9 √x − 3 / x − 9
Textbook Question
Analyze the following limits and find the vertical asymptotes of f(x) = (x − 5) / (x2 − 25).
lim x → -5- f(x)
Textbook Question
Sketch the graph of a function with the given properties. You do not need to find a formula for the function.
f(2) = 1,lim x→2 f(x) = 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)=√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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