Q. 71 - Calculus

Question

Prove each of the limit statements in Exercises 67–72. You

will have to bound $\delta$.

$\underset{x\to 2}{\lim }\frac{4}{x^2}=1$

Step-by-Step Solution

Answer

The given limit $\underset{x\to 2}{\lim }\frac{4}{x^2}=1$ is proved. 

Step 1. Given information.

We are given, 

$\underset{x\to 2}{\lim }\frac{4}{x^2}=1$

Step 2. Proving the limit

Given $\epsilon >0$, choose $\delta =\min \left(1,\frac{\epsilon }{3}\right)$.

Then if $0<|x-2|<\delta$, we have,

$$\begin{gathered} \left|\left(\frac{4}{x^2}\right)-1\right|=\left|\frac{4-x^2}{x^2}\right| \\ =\frac{\left|\right(2-x\left)\right(2+x\left)\right|}{x^2} \\ =|x-2|\frac{|x+2|}{x^2} \\ <\delta \frac{|x+2|}{x^2} \\ \le \delta \left(3\right) \\ \le \left(\frac{\epsilon }{3}\right)3 \\ =\epsilon \end{gathered}$$

Hence Proved.