Textbook Question
Estimate the following limits using graphs or tables.
lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4
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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 2.5.7
Determine the following limits at infinity.
lim t→∞ (−12t^−5)
Verified step by step guidance1
Identify the expression: \(-12t^{-5}\).
Recognize that as \(t\) approaches infinity, the term \(t^{-5}\) represents \(\frac{1}{t^5}\).
Understand that \(\frac{1}{t^5}\) approaches 0 as \(t\) approaches infinity because the denominator grows very large.
Multiply the limit of \(t^{-5}\) by \(-12\), which is a constant, to find the limit of the entire expression.
Conclude that the limit of \(-12t^{-5}\) as \(t\) approaches infinity is 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits at Infinity
Limits at infinity refer to the behavior of a function as the input approaches infinity. This concept is crucial in calculus for understanding how functions behave in extreme cases, particularly for rational functions, polynomials, and exponential functions. Evaluating limits at infinity helps determine horizontal asymptotes and the end behavior of functions.
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One-Sided Limits
Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For example, t^−5 is equivalent to 1/t^5. Understanding negative exponents is essential for simplifying expressions and evaluating limits, especially as the variable approaches infinity, where terms with negative exponents tend to zero.
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6:37Zero and Negative Rules
Behavior of Functions as t Approaches Infinity
The behavior of functions as t approaches infinity involves analyzing how the function's value changes as t increases without bound. In the case of polynomial functions with negative exponents, the function's value will approach zero. This concept is vital for determining the limit of functions at infinity and understanding their long-term trends.
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Related Practice
Textbook Question
Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.
f(x) = x^2−25 / x−5; a=5
290
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Textbook Question
Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.
f(x) = {x^2+1 if x≤−1
3 if x>−1; a=−1
423
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Textbook Question
Determine the following limits at infinity.
lim x→∞ (5 + 1/x +10/x^2)
434
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Textbook Question
Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>
279
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Textbook Question
Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.
g(θ)=tan πθ/10
316
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