Find each limit, if it exists. + 2(а) limx³x? + 2x2 + 2(b) lim*o x? - 1x² + 2(c) lim* 0 X - 1

Evaluate: 1. limx→∞ x2+2x3-12limx→∞ x2+2x2-13. limx→∞ x2+2x-1

Answered: + 2 (а) lim x³ x? + 2 x2 + 2 (b) lim *o…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 7E

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Question

Find each limit, if it exists.

+ 2
(а) lim
x³
x? + 2
x2 + 2
(b) lim
*o x? - 1
x² + 2
(c) lim
* 0 X - 1

Expert Solution

Step 1:To determine

Evaluate:

$1 . \lim_{x \to \infty} \frac{x^{2} + 2}{x^{3} - 1} 2 \lim_{x \to \infty} \frac{x^{2} + 2}{x^{2} - 1} 3 . \lim_{x \to \infty} \frac{x^{2} + 2}{x - 1}$

Step 2:Calculation

1. Consider $\lim_{x \to \infty} \frac{x^{2} + 2}{x^{3} - 1}$

Dividing numerator and denominator by $x^{3}$ , we get,

$\lim_{x \to \infty} \frac{\frac{x^{2} + 2}{x^{3}}}{\frac{x^{3} - 1}{x^{3}}} = \lim_{x \to \infty} \frac{\frac{1}{x} + \frac{2}{x^{3}}}{1 - \frac{1}{x^{3}}} = \frac{\lim_{x \to \infty} \frac{1}{x} + \frac{2}{x^{3}}}{\lim_{x \to \infty} 1 - \frac{1}{x^{3}}} = \frac{\frac{1}{\infty} + \frac{2}{\infty}}{1 - \frac{1}{\infty}} = \frac{0}{1 - 0} = 0$

Hence, $\lim_{x \to \infty} \frac{x^{2} + 2}{x^{3} - 1} = 0$

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+ 2 (а) lim x³ x? + 2 x2 + 2 (b) lim *o x? - 1 x² + 2 (c) lim * 0 X - 1

+ 2 (а) lim x³ x? + 2 x2 + 2 (b) lim o x? - 1 x² + 2 (c) lim 0 X - 1