using delta-epsilon method

Question

lim(x-->3) (x+1)/(x-1)=2

Answer 1

We  want to use the ε - δ definition, we must do is to prove that, for any ε > 0, there is a δ (which usually depends on the ε), such that:

0 < |x -3| < δ ⇒ |(x+1)/(x -1) -2| < ε

This is usually done by starting from the consequent:

|{(x+1) - 2( x - 1)}/(x -1)| = |(x+1 -2x+2)/(x - 1)|  = |( - x + 3)/(x - 1)|

Now  that |- x + 3| is smaller than ε if |x - 1| is smaller than ε, δ = ε.