# Bill's question at Yahoo! Answers regarding the epsilon-delta definition of a limit

#### MarkFL

Staff member
Here is the question:

Calc: epsilon/delta proof?

Write out an epsilon/delta proof to show that:

lim (-3x+1)=-5
x->2
I have posted a link there to this topic so the OP may see my work.

#### MarkFL

Staff member
Hello Bill:

We are given to prove:

$$\displaystyle \lim_{x\to2}(-3x+1)=-5$$

For any given $\epsilon>0$, we wish to find a $\delta$ so that:

$|-3x+1+5|<\epsilon$ whenever $0<|x-2|<\delta$

To do this, consider:

$$\displaystyle |-3x+1+5|=|-3x+6|=3|x-2|$$

Thus, to make:

$$\displaystyle 3|x-2|<\epsilon$$

we need only make:

$$\displaystyle 0<|x-2|<\frac{\epsilon}{3}$$

We may then choose:

$$\displaystyle \delta=\frac{\epsilon}{3}$$

Verification:

If $$\displaystyle 0<|x-2|<\frac{\epsilon}{3}$$, then $$\displaystyle 3|x-2|<\epsilon$$ implies:

$$\displaystyle |3x-6|=|-3x+6|=|(-3x+1)-(-5)|<\epsilon$$