The Student Room Group
Uni students - Complete our survey >>
Uni students - Complete our survey >>
Uni students - Complete our survey >>

What am I doing wrong?

I need show that f(x)= x^2 +2 is continuous at x=3.
So I'm trying to prove that as x approaches 3 from the left = the value as x approaches 3 from the right.
(edited 8 years ago)
image.jpg185.1KB
Reply 1
Avatar for Vorsah
Vorsah
OP
5
The other one is upside down
image.jpg176.4KB
Original post by Vorsah
The other one is upside down

Obviously that approach will work if the left and right hand limits are equal to each other AND equal to the value of the function at that point. However, that's certainly a more complicated way of doing it.

The definition of continuity states that

lim⁡x→af(x)=f(a)\displaystyle \lim_{x \rightarrow a} f(x) = f(a)

and the epsilon-delta version is

∀ϵ>0:∃δ:∣x−a∣<δ⇒∣f(x)−f(a)∣<ϵ\forall \epsilon > 0: \exists \delta: |x-a| < \delta \Rightarrow |f(x) - f(a)| < \epsilon

So this is what you should be after

∣x−3∣<δ⇒∣x2+2−11∣=∣x2−9∣<ϵ|x-3| < \delta \Rightarrow |x^2 + 2 - 11| = |x^2 - 9| < \epsilon

It should be easy for you to bound that by using the difference of two squares. That will give you delta in terms of epsilon.
(edited 8 years ago)
Original post by Indeterminate
That's not the way to prove continuity. You can have functions whose left and right hand limits agree at points where the "ordinary" limit is something completely different (i.e there's a discontinuity).

The definition of continuity states that

lim⁡x→af(x)=f(a)\displaystyle \lim_{x \rightarrow a} f(x) = f(a)


But that's equivalent to lim⁡x→a+f(x)=f(a)\displaystyle \lim_{x \rightarrow a+} f(x) = f(a) and lim⁡x→a−f(x)=f(a)\displaystyle \lim_{x \rightarrow a-} f(x) = f(a) which is what he is trying to prove, I think.
Original post by atsruser
But that's equivalent to lim⁡x→a+f(x)=f(a)\displaystyle \lim_{x \rightarrow a+} f(x) = f(a) and lim⁡x→a−f(x)=f(a)\displaystyle \lim_{x \rightarrow a-} f(x) = f(a) which is what he is trying to prove, I think.


That's a possibility, though you'd usually expect someone to use the normal definition in order to make life easier for themselves.

EDIT: I've edited my post above to incorporate this.
(edited 8 years ago)

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you