How to prove limits using the formal definition(epsilon-delta)?

Question

I was told to prove the limit of (10-2x)=16, as x=-3. Can you help me solve this?

Answer 1

Defintion:

limit f(x) = L as x-->a

For every E>0 there exists Delta_E such that

L-E < f(x) < L+E when a-Delta_E < x < a+Delta_E

Working backwards:

If

16-E < 10-2x < 16+E

Then

6-E < -2x < 6+E

(1/2)E - 3 > x > (-1/2)E - 3

-3 + (1/2)E > x > -3 - (1/2)E

-3 - (1/2)E < x < -3 + (1/2)E

-3 - E/2 < x < -3 + E/2

=======================================

For E>0, Let Delta_E = E/2

When -3-Delta_E < x < -3+Delta_E

-3-E/2 < x < -3 + E/2

-6 - E < 2x < -6 + E

E+6 > -2x > 6-E

E+16 > 10-2x > 16-E

E+16 > f(x) > 16-E

[end of proof]