Proving limits using the formal definition of limits

Question

Prove the lim x->2 (3x-2) = 4 using the epsilon-delta definition of Formal Definition of the Limit

Answer 1

First, notice that | f(x) - L| in this case is | (3x-2) - 4 | = |3x-6|=3|x-2|

This is convenient, since we are taking the limit as x -> 2, and so |x-2| is the exact quantity "controlled by delta".

The proof:

Let eps > 0, define delta = eps/4, and suppose that |x-2| < delta. Then | (3x-2) - 4 | = | 3x-6 | = 3 |x-2| < 3(eps/4) < eps.

Hence, by definition, lim_{x to 2} 3x-2 = 4