Q. 60 - 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. 

$\underset{x\to \infty }{lim}\ln x=\infty , M=100000, \mathrm{find} \mathrm{smallest} N>0$

Step-by-Step Solution

Answer

The required value of $N=e^{100000}$

Step 1. Given Information

The given function is $f\left(x\right)=\ln x$

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

Then $\ln x\in (M,\infty )$

Now the smallest value of N is given by,

$$\begin{gathered} \ln x=100000 \\ x=e^{100000} \end{gathered}$$