Calculus/Limits

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Expert: - 8/2/2010


Question
Hi,

Can you prove these few problems I have had trouble with please?

1. If given a function f and g, for which the limit as x approaches 0 of g(x) doesn't exist, then lim as x approaches 0 of [f(x) + g(x)] doesn't exist if and only if the lim as x approaches 0 of f(x) does exist?

2. Can you prove the same thing for f(g(x)) instead of [f(x) + g(x)]?

3. If lim as x approaches 0 of f(x) exists and doesn't equal 0. Prove that if lim as x approaches 0 of g(x) doesn't exist then lim f(x)g(x) as x approaches 0 doesn't exist.

Thanks,
Kaylee

Answer
1. Lets say that g(x) = 1/x, so at x=0, this is undefined.

If f(x) = -1/x, then the lim(x->0)f(x) is also undefined.
However, f(x) + g(x) = 0, and the lim(x->0)(f(x)+g(x)) does exist.

If f(x) = 1/x, then lim(x->0)(f(x)+g(x)) is undefined.

If lim(x->0)f(x) is defined, then clearly the lim(x->0)(f(x)+g(x)) is undefined as well.

Thus, if the lim(x->0) is defined, the limit on the sum if undefined.
However, if not defined, the limit on the sum could be defined or not.

Calculus

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