Q. 23 - Calculus

Question

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

 

$\underset{x\to 2}{\lim }\frac{x^2-4}{x-2}=4$

Step-by-Step Solution

Answer

For all epsilon positive, there exists a delta positive such that if, 

$x\in (2-\delta ,2)\cup (2,2+\delta ) \mathrm{Then} \frac{x^2-4}{x-2}\in (4-\mathrm{\epsilon },4+\mathrm{\epsilon })$

The graph is 

Step 1. Given information.

The given expression is $\underset{x\to 2}{\lim }\frac{x^2-4}{x-2}=4$.

Step 2. Explanation.

From the given expression, we have, $c=2, L=4$.

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

For all epsilon positive, there exists a delta positive such that if,

$x\in (2-\delta ,2)\cup (2,2+\delta ) \mathrm{Then} \frac{x^2-4}{x-2}\in (4-\mathrm{\epsilon },4+\mathrm{\epsilon })$

Step 3. Graph the expression.

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