Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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I am stuck on 21.
![### Calculus: Limits at Infinity
#### Problems:
**13.** \(\lim_{{x \to +\infty}} \frac{3x + 1}{2x - 5} \)
**15.** \(\lim_{{y \to -\infty}} \frac{3}{y + 4} \)
**17.** \(\lim_{{x \to -\infty}} \frac{x - 2}{x^2 + 2x + 1} \)
**19.** \(\lim_{{x \to +\infty}} \frac{7 - 6x^5}{x + 3} \)
**21.** \(\lim_{{t \to -\infty}} \frac{6 - t^3}{7t^3 + 3} \)
**23.** \(\lim_{{x \to +\infty}} \sqrt[3]{\frac{2 + 3x - 5x^2}{1 + 8x^2}} \)
**25.** \(\lim_{{x \to -\infty}} \frac{\sqrt{5x^2 - 2}}{x + 3} \)
#### Explanation:
These problems involve finding the limit of different rational and radical functions as the variable approaches either positive or negative infinity. Rational functions are fractions involving polynomials, whereas the radical function incorporates a root.
For each problem, the behavior of the function as the variable \( x \) (or \( y \) or \( t \)) grows without bound, either positively or negatively, will be considered. Techniques such as algebraic manipulation and analyzing the highest degree terms in the numerator and denominator will be applied.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f605619-bbba-4f43-942e-b505f4944796%2F2c9cd366-2db5-4a73-a840-68ac2018cf72%2Fdoc2khb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus: Limits at Infinity
#### Problems:
**13.** \(\lim_{{x \to +\infty}} \frac{3x + 1}{2x - 5} \)
**15.** \(\lim_{{y \to -\infty}} \frac{3}{y + 4} \)
**17.** \(\lim_{{x \to -\infty}} \frac{x - 2}{x^2 + 2x + 1} \)
**19.** \(\lim_{{x \to +\infty}} \frac{7 - 6x^5}{x + 3} \)
**21.** \(\lim_{{t \to -\infty}} \frac{6 - t^3}{7t^3 + 3} \)
**23.** \(\lim_{{x \to +\infty}} \sqrt[3]{\frac{2 + 3x - 5x^2}{1 + 8x^2}} \)
**25.** \(\lim_{{x \to -\infty}} \frac{\sqrt{5x^2 - 2}}{x + 3} \)
#### Explanation:
These problems involve finding the limit of different rational and radical functions as the variable approaches either positive or negative infinity. Rational functions are fractions involving polynomials, whereas the radical function incorporates a root.
For each problem, the behavior of the function as the variable \( x \) (or \( y \) or \( t \)) grows without bound, either positively or negatively, will be considered. Techniques such as algebraic manipulation and analyzing the highest degree terms in the numerator and denominator will be applied.
![### Calculus - Limits
#### Problem 19
Evaluate the limit:
\[ \lim_{x \to +\infty} \frac{7 - 6x^5}{x + 3} \]
Solution:
\[ \lim_{x \to +\infty} \frac{7 - 6x^5}{x + 3} = \lim_{x \to +\infty} \frac{-6x^5 + 7}{x + 3} \]
Since the highest power term in the numerator is \( -6x^5 \) and in the denominator is \( x \), the limit tends towards:
\[ -\infty \]
because the top value (numerator) is negative.
#### Problem 21
Evaluate the limit:
\[ \lim_{t \to +\infty} \frac{6 - t^3}{7t^3 + 3} \]
Solution:
\[ \lim_{t \to +\infty} \frac{6 - t^3}{7t^3 + 3} = \lim_{t \to +\infty} \frac{-t^3 + 6}{7t^3 + 3} \]
Divide the numerator and the denominator by \( t^3 \):
\[ = \lim_{t \to +\infty} \frac{-1 + \frac{6}{t^3}}{7 + \frac{3}{t^3}} \]
As \( t \to +\infty \), \(\frac{6}{t^3}\) and \(\frac{3}{t^3}\) approach 0:
\[ = \frac{-1 + 0}{7 + 0} = \frac{-1}{7} = -\frac{1}{7} \]
#### Problem 23
Evaluate the limit:
\[ \lim_{x \to -\infty} \sqrt[3]{\frac{2 + 3x - 5x^2}{1 + 9x^2}} \]
Solution:
The highest degrees in the numerator and denominator are \( x^2 \). Divide by \( x^2 \):
\[ \lim_{x \to -\infty} \sqrt[3]{\frac{\frac{2}{x^2} + \frac{3}{x} - 5}{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f605619-bbba-4f43-942e-b505f4944796%2F2c9cd366-2db5-4a73-a840-68ac2018cf72%2Fjg08dk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus - Limits
#### Problem 19
Evaluate the limit:
\[ \lim_{x \to +\infty} \frac{7 - 6x^5}{x + 3} \]
Solution:
\[ \lim_{x \to +\infty} \frac{7 - 6x^5}{x + 3} = \lim_{x \to +\infty} \frac{-6x^5 + 7}{x + 3} \]
Since the highest power term in the numerator is \( -6x^5 \) and in the denominator is \( x \), the limit tends towards:
\[ -\infty \]
because the top value (numerator) is negative.
#### Problem 21
Evaluate the limit:
\[ \lim_{t \to +\infty} \frac{6 - t^3}{7t^3 + 3} \]
Solution:
\[ \lim_{t \to +\infty} \frac{6 - t^3}{7t^3 + 3} = \lim_{t \to +\infty} \frac{-t^3 + 6}{7t^3 + 3} \]
Divide the numerator and the denominator by \( t^3 \):
\[ = \lim_{t \to +\infty} \frac{-1 + \frac{6}{t^3}}{7 + \frac{3}{t^3}} \]
As \( t \to +\infty \), \(\frac{6}{t^3}\) and \(\frac{3}{t^3}\) approach 0:
\[ = \frac{-1 + 0}{7 + 0} = \frac{-1}{7} = -\frac{1}{7} \]
#### Problem 23
Evaluate the limit:
\[ \lim_{x \to -\infty} \sqrt[3]{\frac{2 + 3x - 5x^2}{1 + 9x^2}} \]
Solution:
The highest degrees in the numerator and denominator are \( x^2 \). Divide by \( x^2 \):
\[ \lim_{x \to -\infty} \sqrt[3]{\frac{\frac{2}{x^2} + \frac{3}{x} - 5}{\
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