Q. 32 - 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.

$\underset{x\to -\infty }{\lim }\frac{1}{x^2+1}=0$

Step-by-Step Solution

Answer

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

$x\in (-\infty ,N) \mathrm{Then} \frac{1}{x^2+1}\in (0-\mathrm{\epsilon },0+\mathrm{\epsilon })$.

The graph is

Step 1. Given information.

The given expression is$\underset{x\to -\infty }{\lim }\frac{1}{x^2+1}=0$.

Step 2. Explanation.

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

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

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

$x\in (-\infty ,N) \mathrm{Then} \frac{1}{x^2+1}\in (0-\mathrm{\epsilon },0+\mathrm{\epsilon })$.

Step 3. Graph the expression.

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