Calculus: Epsilon Delta Question

Question

Define a function f by f(x)=2x−1. Find the largest number δ>0 such that, if |x−1|<δ then |f(x)−1|<12. Enter your answer as a decimal number.

Answer 1

|(2x−1)-1|=|2x-2|=|2(x-1)|=2|x-1|    but |x-1|<δ   so

|(2x−1)-1|=2|x-1|<2δ=12  so  δ=6

 

Note: You can check this since |x-1|<6 -->  -6<x-1<6  -->  -5<x<7

and f(7)=2(7)-1=13  so  |f(7)-1|=12  etc.