Textbook Question
Determine the interval(s) on which the following functions are continuous; then analyze the given limits.
f(x)=csc x;lim x→π/4f (x);lim x→2π^− f(x)
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 65
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim t→2+ |2t − 4|t^2 − 4
Verified step by step guidance1
Identify the type of limit: This is a one-sided limit as \( t \to 2^+ \).
Simplify the expression inside the absolute value: \(|2t - 4| = |2(t - 2)| = 2|t - 2|\).
Substitute \( t = 2^+ \) into the expression \( |2t - 4| \). Since \( t \to 2^+ \), \( t - 2 \to 0^+ \), so \(|t - 2| = t - 2\).
Rewrite the limit expression: \( \lim_{t \to 2^+} \frac{2(t - 2)}{t^2 - 4} \).
Factor the denominator: \( t^2 - 4 = (t - 2)(t + 2) \), and simplify the expression to \( \lim_{t \to 2^+} \frac{2}{t + 2} \).
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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 specific points, including points of discontinuity. In this question, we are interested in the limit as t approaches 2 from the right (2+), which requires evaluating the function's behavior just above this point.
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One-Sided Limits
Absolute Value Function
The absolute value function, denoted as |x|, represents the distance of x from zero on the number line, effectively removing any negative sign. In the context of the limit problem, |2t - 4| will change its expression depending on whether 2t - 4 is positive or negative. Understanding how to handle absolute values is crucial for correctly evaluating the limit as t approaches 2.
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Indeterminate Forms
Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In this problem, the expression |2t - 4|/(t^2 - 4) may lead to an indeterminate form as t approaches 2, necessitating further analysis, such as factoring or applying L'Hôpital's Rule to resolve the limit.
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Related Practice
Textbook Question
Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).
f(x) = (x2 − 4x + 3) / (x − 1)
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.
If limx→a f(x) = L, then f(a)=L.
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Textbook Question
Determine the interval(s) on which the following functions are continuous; then analyze the given limits.
f(x)=1+sin x / cos x; limx→π/2^− f(x); lim x→4π/3 f(x)
Textbook Question
Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (x2 − 4x + 3) / (x − 1)
Textbook Question
Evaluate each limit.
lim x→e^2 ln^2x−5 ln x+6 lnx−2