Q. 39 - 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.

$\underset{x\to -\infty }{\lim }f\left(x\right)=L$

Step-by-Step Solution

Answer

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

$x\in \left(-\infty ,N\right) \mathrm{then} f\left(x\right)\in \left(L-\epsilon ,L+\epsilon \right)$.

The graph is 

Step 1. Given information.

The given expression is $\underset{x\to -\infty }{\lim }f\left(x\right)=L$.

Step 2. Explanation.

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

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

For all $\epsilon >0$, there is some $N<0$ such that if,

$x\in \left(-\infty ,N\right) \mathrm{then} f\left(x\right)\in \left(L-\epsilon ,L+\epsilon \right)$.

Step 3. Graph the expression.

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