Chapter 1: Q. 87 (page 151)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information

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

Step 2. Proving the statement

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

If $δ = \min (δ_{1}, δ_{2})$ , then whenever $x \in (c - δ, c) \cup (c, c + δ)$ we also have,

$L - ε < l (x) \leq f (x) \leq u (x) < L + ε$

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

$\lim_{x \to c} \frac{f (x)}{g (x)} = - \infty$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Probability and Statistics

Applied Mathematics

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Probability and Statistics

Applied Mathematics

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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