- #1
tanzl
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Homework Statement
Prove that lim (f(x) , x=a) = a4 where f(x)=x4
Homework Equations
I think I understand the mechanics of limit proof. I just want to improve my way of presenting it because sometimes even I feel ambiguous for my own works.
The Attempt at a Solution
For all [tex]\epsilon[/tex]>0, there exists [tex]\delta[/tex]>0 such that if 0 < |x-a| < [tex]\delta[/tex], then |f(x)-a4| < [tex]\epsilon[/tex]
I think the mechanic of the proof is I need to find delta in terms of epsilon such that it satisfies all the conditions above. (or another way round, for me, the proof more like I need to express f(x)-L where L is the limit in terms of delta so that I could make it smaller than epsilon... please correct me)
So, the first thing I need to do is to express |f(x)-a4| in terms of delta.
|f(x)-a4|
=|x^4 - a^4|
=|x-a| * |x3 + ax2 + a2x + a3|
<=|x-a| * |x3| + |ax2| + |a2x| + |a3|
My problem comes, (when it starts to sound ambiguous)
What I do is I will let |x-a| < [tex]\delta[/tex]1 where [tex]\delta[/tex]1 is a positive number such that |x3| + |ax2| + |a2x| + |a3| < P where P is a positive number.
My reason of doing this is I need to obtain a close interval for the expression |x3| + |ax2| + |a2x| + |a3| so that I can make |f(x) - L| smaller than epsilon. Therefore, I started out by letting |x-a| smaller than [tex]\delta[/tex]1 which is a arbitrary positive number. Then by algebraic manipulation, I can obtain an inequality of |x| smaller than [tex]\delta[/tex]1 +a . So when I substitute [tex]\delta[/tex]1 +a into each terms of |x3| + |ax2| + |a2x| + |a3|, I can get inequality of each terms smaller than something. To save the trouble, I let the whole expression less than P which is also an arbitrary positive number.
Since |x3| + |ax2| + |a2x| + |a3| < P
then |x-a| * |x3| + |ax2| + |a2x| + |a3| < [tex]\delta[/tex] *P
So, when I let [tex]\delta[/tex] *P = [tex]\epsilon[/tex]. (I should express delta in terms of epsilon or epsilon in terms of delta?? The first sounds more correct for me because our mission is to find a delta for any given epsilon)
|f(x)-a4| < [tex]\epsilon[/tex]
Done.
Please feel free to correct me.