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