Saturday, May 26, 2018

Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 19

Show that the statement limx2(145x)=4 using the precise definition of a limit.

Based from the definition,

if 0<|xa|<δ then |f(x)L|<ε

if 0<|x2|<δ then |(145x)4|<ε

But,

|(145x)4|=|145x4|=|105x|=|5(x2)|=5|x2|

So we want

if 0<|x2|<δ then 5|x2|<ε

That is,

if 0<|x2|<δ then |x2|<ε5

The statement suggest that we should choose δ=ε5.

By proving that the assumed value of δ=ε5 will fit the definition.

if 0<|x2|<δ then,

|(145x)4|=|145x4|=|105x|=|5(x2)|=5|x2|<5δ=\cancel5(ε\cancel5)=ε

Thus,

if 0<|x2|<δ then |(145x)4|<ε

Therefore, by the precise definition of a limit

limx2(145x)=4

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