Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Deduce???

I am puzzled. What does the question want me to do when it tells me to "deduce"?

i) Evaluate the integral:

in the cases k ≠ 0 and k = 0.

ii) Deduce that

The above are the results when the two said cases are considered. Obviously, as k gets closer to 0, the two functions being integrated become increasingly similar, and would result therefore in a similar integral. What exactly does the second part of the question want me to do?

Thanks.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Deduce???

"Deduce" means "Hence prove", or "Therefore, prove that"

k = 0:

k ≠ 0

Let u = x+1, du = dx:

Therefore, as

is a continuous function, it must converge to

as k moves to zero from either side.

Last edited by Identity (2008-03-30 11:00:59)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Deduce???

All it means is to use the first result to prove (obtain, show, ...) the second.

Identity, there is no concept of continuity for a class of functions.  There is equicontinuous which is somewhat similar, but I don't believe that's what you meant.

Since it's freshman calculus, I would just make a statement similar to Identity's (but without the word "continuity" ).  If it was analysis, you would need a rigorous proof.  The proof would go something like this.

Pick {k_n} to be a sequence of real numbers such that k_i < k_{i-1} for all i and k_n -> 0 as n -> infinity.  Now prove that this sequence of functions uniformly converges to 1/(x+1).  Then there is a theorem that says:

And you're done.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Deduce???

This method actually looks really cool, can it be used for many other limits?