Q. 19 - 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.

$\underset{x\to -3}{\lim }\sqrt{x+7}=2$

Step-by-Step Solution

Answer

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

$x\in (-3-\delta ,-3)\cup (-3,-3+\delta ) \mathrm{Then}\mathit{ }\sqrt{x\mathit{+}\mathit{7}}\in (2-\mathrm{\epsilon },2+\mathrm{\epsilon })$.

The graph is 

Step 1. Given information.

The given expression is $\underset{x\to -3}{\lim }\sqrt{x+7}=2$.

Step 2. Explanation.

From the given expression, we have,  $c=-3, L=2$.

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

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

$x\in (-3-\delta ,-3)\cup (-3,-3+\delta ) \mathrm{Then}\mathit{ }\sqrt{x\mathit{+}\mathit{7}}\in (2-\mathrm{\epsilon },2+\mathrm{\epsilon })$.

Step 3. Graph the expression.

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