Q. 63 - Calculus

Question

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N  that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.    

$\underset{x\to \infty }{lim}(4-x^2)=-\infty , M=-100, \mathrm{find} \mathrm{smallest} N>0$

Step-by-Step Solution

Answer

The required value of $N\approx 10.198$

Step 1. Given Information

The given function is $f\left(x\right)=4-x^2$

Step 2. Explanation

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

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

For all M negative, there is some N positive such that if $x\in (N,\infty )$

Then $f\left(x\right)\in (-\infty ,M)$

Now the smallest value of N is given by,

$$\begin{gathered} 4-x^2=-100 \\ {x}^2=104 \\ x=\sqrt{104} \\ x\approx 10.198 \end{gathered}$$