- #1
Saladsamurai
- 3,020
- 7
Problem
I've just kind of been 'going through the motions,' but I feel like the Precise Definition of a Limit is about to set in.
I think some questions about this example should help.
Prove the Limit Statement:
[tex]\lim_{x\rightarrow4}(9-x)=5[/tex]
Attempt
So by asserting that the limit is indeed '5' we are implying that there exists some [itex]\delta[/itex] such that for all 'x'
[itex]0<|x-x_o|<\delta\Rightarrow|f(x)-5|<\epsilon[/itex]
So:
[itex]\-\epsilon<(9-x)-5<\epsilon[/itex]
[itex]-\epsilon-4<-x<\epsilon-4[/itex]
[itex]4-\epsilon<x<\epsilon+4[/itex]
[itex]\therefore[/itex]
[itex]\, -\epsilon<x-4<\epsilon[/itex]
[itex]|x-4|<\epsilon=\delta[/itex]
Now I am a little confused. Have I actually done anything?
Have I shown that so long as I stay within [itex]\delta=\epsilon \text{ of }x_o[/itex] I can get within a distance of [itex]\epsilon[/itex] of 'L.'
Because that's what i am under the impression I have done. But I am not confident about it.
Thanks