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→2 (5x−6)^3/2
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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 32
Determine the following limits.
lim x→∞ 6x2/(4x^2+√(16x4 + x2))
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Identify the dominant terms in the numerator and the denominator. In the numerator, the dominant term is \(6x^2\). In the denominator, the dominant term is \(4x^2\) and the term under the square root is \(16x^4\).
Factor out the highest power of \(x\) from both the numerator and the denominator. In this case, factor \(x^2\) from both.
Rewrite the expression: \(\frac{6x^2}{4x^2 + \sqrt{16x^4 + x^2}} = \frac{6}{4 + \sqrt{16 + \frac{1}{x^2}}}\).
As \(x\) approaches infinity, the term \(\frac{1}{x^2}\) approaches zero. Simplify the expression to \(\frac{6}{4 + \sqrt{16}}\).
Calculate the simplified expression by evaluating the square root and performing the arithmetic operations.
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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 involve evaluating the behavior of a function as the variable approaches infinity. This concept is crucial for understanding how functions behave for very large values, often simplifying expressions by focusing on the highest degree terms in polynomials or rational functions.
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One-Sided Limits
Dominant Terms
In the context of limits, dominant terms are the terms in a polynomial or rational expression that have the highest degree and thus dictate the behavior of the function as the variable approaches infinity. Identifying these terms allows for simplification of the limit calculation, as lower degree terms become negligible.
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2:02Simplifying Trig Expressions Example 1
Rational Functions
A rational function is a ratio of two polynomials. Understanding the properties of rational functions, including their limits, is essential for solving limit problems. The behavior of rational functions at infinity often depends on the degrees of the numerator and denominator, which can lead to finite limits, infinite limits, or limits that do not exist.
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6:04Intro to Rational Functions
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 x^2 − 1 / x − 1
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Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
a. The value of does not exist.
Textbook Question
Determine the following limits.
lim x→3- x − 4 / x^2 − 3x
Textbook Question
Determine the following limits.
lim x→−∞ 40x^4+x^2+5x / √64x^8+x^6
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Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
d. . (Hint: Graph y=√x)