Q. 62 - 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.   

$\underset{x\to -\infty }{lim}{3}^x=0, \epsilon =\frac{1}{4}, \mathrm{find} \mathrm{smallest}-\mathrm{magnitude} N<0$

Step-by-Step Solution

Answer

The required value of $N\approx -1.26186$

Step 1. Given Information

The given function is $f\left(x\right)=3^x$

Step 2. Explanation

From the given function, we have, $c=-\infty ,L=0$

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

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

Then $3^x\in (0-\epsilon ,0+\epsilon )$

Now the smallest value of N is given by,

$$\begin{gathered} 3^x=0+\frac{1}{4} \\ \ln 3^x=\ln \left(\frac{1}{4}\right) \\ x\ln 3=\ln \left(\frac{1}{4}\right) \\ x=\frac{\ln \left({\displaystyle \frac{1}{4}}\right)}{\ln 3} \\ x\approx \frac{-1.3863}{1.0986} \\ x\approx -1.26186 \end{gathered}$$