How do you use the epsilon delta definition to prove that the limit of (1+x)/(4+x)=1/21+x4+x=12 as x->2x2?

1 Answer
Feb 23, 2017

See below.

Explanation:

Given a positive epsilonε, let delta = min{1, 14epsilon}. Note that delta is positive.

For any x with 0 < abs(x-2) < delta, we have

abs(x-2) < 1, which is equivalent to -1 < x-2 < 1, and this entails that -5 < x+4 < 7, so we will have -10 < 2(x+4) < 14 and, finally we are assured that abs(2(x+4)) < 14

So for any x with 0 < abs(x-2) < delta, we have

abs((1+x)/(4+x) - 1/2) = abs((2+2x-4-x)/(2(x+4)))

= abs((x-2)/(2(x+4))

< (abs(x-2))/14

< delta/14

< (14epsilon)/14

= epsilon

We have shown that for any epsilon > 0 there is a delta > 0 such that

for any x with 0 < abs(x-2) < delta

abs((1+x)/(4+x) - 1/2)< epsilon