Q. 24 - 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.

 

$\underset{x\to 1}{\lim }\left(x^2-3\right)=-2$

Step-by-Step Solution

Answer

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

$x\in (1-\delta ,1)\cup (1,1+\delta ) \mathrm{Then} \left(x^2-3\right)\in (-2-\mathrm{\epsilon },-2+\mathrm{\epsilon })$.

The graph is

Step 1. Given information.

The given expression is $\underset{x\to 1}{\lim }\left(x^2-3\right)=-2$.

Step 2. Explanation.

From the given expression, we have,$c=1, L=-2$.

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

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

$x\in (1-\delta ,1)\cup (1,1+\delta ) \mathrm{Then} \left(x^2-3\right)\in (-2-\mathrm{\epsilon },-2+\mathrm{\epsilon })$

Step 3. Graph the expression.

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