Q. 4 - Calculus

Question

Suppose you show that $\left|\right(1-2x)-(-5\left)\right|<0.05$ for all x with $0<|x-3|<0.025$Explain why this does not prove that  $\underset{x\to 3}{lim}(1-2x)=-5$

Step-by-Step Solution

Answer

As $-\frac{\epsilon }{2}<|x-3|<\delta$ thus we cannot say that $\underset{x\to 3}{lim}(1-2x)=-5$

Step 1. Given Information

The given limit expression is $\underset{x\to 3}{lim}(1-2x)=-5$

Step 2. Explanation

From the given limit expression, we have,

$f\left(x\right)=1-2x, c=3, L=-5$

Hence, delta-epsilon statement will be as follows,

For all epsilon positive there exists a delta positive if $0<|x-3|<\delta$

Then,

$$\begin{gathered} \left|\right(1-2x)-(-5\left)\right|<\epsilon \\ |1-2x+5|<\epsilon \\ |6-2x|<\epsilon \\ -2|x-3|<\epsilon \\ |x-3|>-\frac{\epsilon }{2} \end{gathered}$$