Q. 30 - Calculus

Question

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$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.

$\underset{x\to 0^{+}}{\lim }\frac{1}{1-e^x}=-\infty$

Step-by-Step Solution

Answer

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

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

The graph is  

Step 1. Given information.

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

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,

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

Step 3. Graph the expression.

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