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Chapter 1: Q. 38 (page 135)

Calculate each of the limits in Exercises $29 - 70$ .
$\lim_{x 鈫� - 3} \frac{3 x^{2} + 8 x - 3}{x + 3}$ .

Short Answer

Expert verified

The value of $\lim_{x 鈫� - 3} \frac{3 x^{2} + 8 x - 3}{x + 3}$ is $- 10$ .

Step by step solution

01

Step 1. Given information

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

02

Step 2. The limit is calculated as below:

$\lim_{x 鈫� - 3} \frac{3 x^{2} + 8 x - 3}{x + 3} = \lim_{x 鈫� - 3} \frac{3 x^{2} + 9 x - x - 3}{x + 3} = \lim_{x 鈫� - 3} \frac{3 x (x + 3) - 1 (x + 3)}{x + 3} = \lim_{x 鈫� - 3} \frac{(x + 3) (3 x - 1)}{(x + 3)} = \lim_{x 鈫� - 3} (3 x - 1)$

$= \lim_{x 鈫� - 3} (3 x) - \lim_{x 鈫� - 3} (1)$ [Sum rule]

$= 3 (- 3) - 1 = - 9 - 1 = - 10$

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Most popular questions from this chapter

Write each of the inequalities in interval notation:

$0 < | x + 3 | < 0.05$

For each limit $\underset{x 鈫� c}{l i m} f (x) = L$ in Exercises 43鈥�54, use graphs and algebra to approximate the largest value of $未$ such that if $x 鈭� (c - 未, c) 鈭� (c, c + 未) then f (x) 鈭� (L - 蔚, L + 蔚)$

$\underset{x 鈫� - 1}{l i m} \frac{x^{2} - 2 x - 3}{x + 1} = - 4, 蔚 = 0.1$

Write delta-epsilon proofs for each of the limit statements $\lim_{x 鈫� c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x 鈫� 0} (3 x^{2} + 1) = 1$ .

Write each of the inequalities in interval notation:

$|f (x) - L| < 鈭�$

In Exercises 39鈥�44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f (x) = \{\begin{cases} (x - 3), i f x < 3 \\ - (x - 3), i f x 猢� 3 \end{cases} .$

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