Deduce that if |x -3| < 1, then |(x + 2)^(2) - 25| < 11|x - 3|
Use this result to show, without using the Limit Laws, that
lim x -> 3 of (x + 2)^(2) = 25
I am really confused and dont understand this question
$\displaystyle |x-3|<1 $
$\displaystyle |(x+2)^2 - 25| = |(x+2)^2 - 5^2| = |(x+2-5)(x+2+5)|$
$\displaystyle = |(x-3)(x+7)| $
Also from $\displaystyle |x-3|<1 $ , we have $\displaystyle 2<x<4 $ so $\displaystyle 9 < x+7 < 11 $
Therefore , $\displaystyle |(x+2)^2 - 25| = |x-3||x+7| < 11|x-3|$
For the second part , we have to prove it by checking the definition ... What similarity between part 1 and the definition can you find ?