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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 14: Q. 10 (page 905)

find the limit $\lim_{x 鈫� 2} \frac{x^{3} - 8}{x^{3} - 2 x^{2} + 4 x - 8}$

Short Answer

Expert verified

$\lim_{x 鈫� 2} \frac{x^{3} - 8}{x^{3} - 2 x^{2} + 4 x - 8} = \frac{3}{2}$

Step by step solution

01

Step 1. Given information

$\lim_{x 鈫� 2} \frac{x^{3} - 8}{x^{3} - 2 x^{2} + 4 x - 8}$

02

Step 2. limit of the function

We can find limit of rational function by $\lim_{x 鈫� c} f (x) = f (c)$ if c is in the domain of f(x) and in the domain of f(x)= $\frac{x^{3} - 8}{x^{3} - 2 x^{2} + 4 x - 8}$ 2 is not included. So we have to simplify the f(x) for finding the limit at 2.

$\lim_{x 鈫� 2} \frac{x^{3} - 8}{x^{3} - 2 x^{2} + 4 x - 8} = \lim_{x 鈫� 2} \frac{(x - 2) (x^{2} + 2 x + 4)}{(x^{2} + 4) (x - 2)} = \lim_{x 鈫� 2} \frac{x^{2} + 2 x + 4}{x^{2} + 4} = \frac{(2)^{2} + 2 (2) + 4}{(2)^{2} + 4} = \frac{3}{2}$

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