Q. 40 - 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. 

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

Step-by-Step Solution

Answer

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

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

The graph is

Step 1. Given information.

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

Step 2. Explanation.

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

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

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

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

Step 3. Graph the expression.

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