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finding mathematical limits?
Please help with maths question...
let f(x) = [((a+b)*x + (a-b)|x|) / 2x ]
all values of a, b such that lim x->0 f(x) exists
I think the answer is a = b, but does this mean all values are possible? ie -inf to +inf ?
2 Answers
- Anonymous1 decade agoFavourite answer
You are right.
If x > 0 then f(x) = [(a + b)x + (a - b)x]/2x = 2ax/2x = a
If x < 0 then f(x) = [(a + b)x + (a - b)(-x)]/2x = 2bx/2x = b
The limit exists if these are the same so a = b is the condition and yes this does mean that a, b can take any value. The only problem might have been if a = b = 0 but this makes f(x) = 0 whether x is positive or negative so the limit still exists.
- Anonymous1 decade ago
There is a more general analysis of this problem:
If a > b there is a one- sided limit as x->0+ and that is a.
If a < b there is a one-sided limit as x->0+ and that is b.
If a > b there is a one- sided limit as x->0- and that is -a.
If a < b there is a one-sided limit as x->0+ and that is -b.
It is only when a = b that the two one-sided limits are the same and thus there is actually a limit and that limit is 0.
Notice if a -> b- or a -> b+ there is a limit of as x->0 and that is 0.
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