Q. 87 Prove the second part of Theorem... [FREE SOLUTION]

Chapter 1: Q. 87 (page 151)

Prove the second part of Theorem $1.29$ : If $\lim_{x 鈫� c} \frac{f (x)}{g (x)}$ is of the form $\frac{1}{0^{-}}$ , then $\lim_{x 鈫� c} \frac{f (x)}{g (x)} = - 鈭�$ .

Short Answer

Expert verified

It is proved that If $\lim_{x 鈫� c} \frac{f (x)}{g (x)}$ is of the form $\frac{1}{0^{-}}$ , then $\lim_{x 鈫� c} \frac{f (x)}{g (x)} = - 鈭�$ .

Step by step solution

Step 1. Given Information

We are given two functions $f (x) and g (x)$ .

Step 2. Proving the statement

Given $蔚 > 0$ , we can choose $未_{1} > 0$ to get $u (x)$ within $蔚$ of L and also choose $未_{2}$ to get $l (x)$ within $蔚$ of L.

If $未 = \min (未_{1}, 未_{2})$ , then whenever $x 鈭� (c - 未, c) 鈭� (c, c + 未)$ we also have,

$L - 蔚 < l (x) 鈮� f (x) 鈮� u (x) < L + 蔚$

Hence, if $\lim_{x 鈫� c} \frac{f (x)}{g (x)}$ is of the form $\frac{1}{0^{-}}$ then the left part will be the result, and it will be,

$\lim_{x 鈫� c} \frac{f (x)}{g (x)} = - 鈭�$ .

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91影视

Prove the second part of Theorem $1.29$ : If $\lim_{x 鈫� c} \frac{f (x)}{g (x)}$ is of the form $\frac{1}{0^{-}}$ , then $\lim_{x 鈫� c} \frac{f (x)}{g (x)} = - 鈭�$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

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91影视

Prove the second part of Theorem 1.29: If limx鈫�cf(x)g(x) is of the form 10-, thenlimx鈫�cf(x)g(x)=-鈭�.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Mechanics Maths

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Prove the second part of Theorem $1.29$ : If $\lim_{x 鈫� c} \frac{f (x)}{g (x)}$ is of the form $\frac{1}{0^{-}}$ , then $\lim_{x 鈫� c} \frac{f (x)}{g (x)} = - 鈭�$ .