Limit formal definition

1. May 26, 2014

TommG

1. The problem statement, all variables and given/known data

Use formal definition of limits
Find L = lim x→ c f(x). Then find a number δ > 0 such for all x

f(x) = 3 - 2x
c = 3
ε = 0.02

3. The attempt at a solution

limx→3 3 -2x

limx→3 3 - limx→3 2x

3 - 2(3) = -3
L = -3

I am not sure how to find delta

2. May 26, 2014

LCKurtz

Ask yourself how close $x$ needs to be to $3$ so that $|f(x)-L|<\epsilon$ or, for your problem, $|(3-2x) - (-3)|<.02$.

3. May 26, 2014

lurflurf

|f(x)-L|<ε
when
|3-x|<δ
write
|f(x)-L|
in terms of
|3-x|

4. May 26, 2014

TommG

ok so i take

-0.02 < (3-2x)-(-3) < 0.02
-0.02 < 6-2x < 0.02
-6.02 < -2x < -5.98
3.01 > x > -2.99
(-2.99,3.01)

-2.99 - 3 = -5.99
3.01 - 3 = 0.1

so since δ > 0
δ = 0.1

5. May 26, 2014

HallsofIvy

Notice that, at this point, you could say
-0.02 < 2(3- x)< 0.02
-0.01< 3- x< 0.01 so that |x- 3|< 0.01

6. May 26, 2014

LCKurtz

That's good. But you could write it much neater:$$|6-2x| <.02$$ $$2|3-x| <.02$$ $$|3-x| <.01$$The steps are reversible so $\delta=.01$.

7. May 26, 2014

LCKurtz

Also note that should be -2.99 - (-3) = .01. And .1 should be .01.

Last edited: May 26, 2014