Q. 72 - Calculus

Question

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.

$\underset{x\to 3}{\lim }\frac{18}{x^2}=2$

Step-by-Step Solution

Answer

$\underset{x\to 3}{\lim }\frac{18}{x^2}=2$

Step 1: Given information

The expression is 

$\underset{x\to 3}{\lim }\frac{18}{x^2}=2$

Step 2: Calculation

From the limit expression,

$$\begin{gathered} f\left(x\right)=\frac{18}{x^2} \\ c=3 \\ L=2 \end{gathered}$$

Given$\epsilon >0,$ choose $\delta =\min \left(1,\frac{\epsilon }{6}\right)$.

Then if $0<|x-3|<\delta$, we have,

$$\begin{gathered} \left|\left(\frac{18}{x^2}\right)-2\right| \\ =\left|\frac{18-2x^2}{x^2}\right| \\ =\frac{2\left|\right(3-x\left)\right(3+x\left)\right|}{x^2} \\ =2|x-3|\frac{|x+3|}{x^2}<2\delta \frac{|x+3|}{x^2} \\ \le 2\delta \left(3\right) \\ \le \left(\frac{\epsilon }{6}\right)6 \\ =\epsilon \end{gathered}$$

Hence, $\underset{x\to 3}{\lim }\frac{18}{x^2}=2.$