Chapter 1: Q. 30 (page 98)

Write each limit in Exercises 19–42 as a formal statement involving $δ, ε, N, and / or M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.
$\lim_{x \to 0^{+}} \frac{1}{1 - e^{x}} = - \infty$

Short Answer

Expert verified

For all $M < 0$ , there exists a delta positive such that if,

$x \in (0, 0 + δ) Then \frac{1}{1 - e^{x}} \in (- \infty, M)$ .

The graph is

Step by step solution

Step 1. Given information.

The given expression is $\lim_{x \to 0^{+}} \frac{1}{1 - e^{x}} = - \infty$ .

Step 2. Explanation.

From the given expression, we have, $c = 0, L = - \infty$ .

The limit expression can be written as a formal statement as below,

For all epsilon positive, there exists a delta positive such that if,

$x \in (0, 0 + δ) Then \frac{1}{1 - e^{x}} \in (- \infty, M)$

Step 3. Graph the expression.

The graph illustrating the roles of the constants is shown as below:

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Write each limit in Exercises 19–42 as a formal statement involving $δ, ε, N, and / or M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.
$\lim_{x \to 0^{+}} \frac{1}{1 - e^{x}} = - \infty$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explanation.

Step 3. Graph the expression.

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Write each limit in Exercises 19–42 as a formal statement involving δ,ε,N,and/orM,and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.limx→0+11-ex=-∞

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explanation.

Step 3. Graph the expression.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Logic and Functions

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Write each limit in Exercises 19–42 as a formal statement involving $δ, ε, N, and / or M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.
$\lim_{x \to 0^{+}} \frac{1}{1 - e^{x}} = - \infty$