Q. 22 - 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.

 

$\underset{x\to 1^{-}}{\lim }\sqrt{1-x}=0$

Step-by-Step Solution

Answer

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

$x\in (1-\delta ,1)\cup (1,1+\delta ) \mathrm{Then} \sqrt{1-x}\in (0-\mathrm{\epsilon },0+\mathrm{\epsilon })$

The graph is 

Step 1. Given information.

The given expression is $\underset{x\to 1^{-}}{\lim }\sqrt{1-x}=0$.

Step 2. Explanation.

given expression, we have, $c=1, L=0$.

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

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

$x\in (1-\delta ,1)\cup (1,1+\delta ) \mathrm{Then} \sqrt{1-x}\in (0-\mathrm{\epsilon },0+\mathrm{\epsilon })$.

Step 3. Graph the expression.

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