Q. 31 - 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.

$\underset{x\to \infty }{\lim }\frac{x}{1-2x}=-5$

Step-by-Step Solution

Answer

For all epsilon positive, there exists $\mathrm{N}>0$ such that if,

$x\in (N,\infty ) \mathrm{Then} \frac{1}{1-2x}\in (-0.5-\mathrm{\epsilon },-0.5+\mathrm{\epsilon })$

The graph is 

Step 1. Given information.

The given expression is$\underset{x\to \infty }{\lim }\frac{x}{1-2x}=-5$.

Step 2. Explanation.

From the given expression, we have,$c=\infty , L=-0.5$.

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

For all epsilon positive, there exists $N>0$ such that if,

$x\in (N,\infty ) \mathrm{Then} \frac{1}{1-2x}\in (-0.5-\mathrm{\epsilon },-0.5+\mathrm{\epsilon })$

Step 3. Graph the expression.

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