Q. 55 - 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.

$\underset{x\to 1^{+}}{lim}\frac{1}{x^2-1}=\infty , M=1000, \mathrm{find} \mathrm{largest} \delta >0$

Step-by-Step Solution

Answer

The required value of $\delta \approx 0.0005$

Step 1. Given Information

The given function is $f\left(x\right)=\frac{1}{x^2-1}$

Step 2. Explanation

From the given function, we have, $c=1,M=1000$

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

For all M positive, there exists a delta positive such that if $x\in (1,1+\delta )$

Then $\frac{1}{x^2-1}\in (1000,\infty )$

Now the largest value of delta is given by,

$$\begin{gathered} \frac{1}{x^2-1}=1000 \\ {x}^2-1=\frac{1}{1000} \\ {x}^2=0.001+1 \\ x=\sqrt{1.001} \end{gathered}$$

Hence,

style="max-width: none; vertical-align: -49px;" $$\begin{gathered} \delta =\sqrt{1.001}-1 \\ \approx 1.000499-1 \\ \approx 0.000499 \\ \approx 0.0005 \end{gathered}$$