mathematics-limits?

Question

lim sqrt(3+2x) -sqrt(2)-1
x->sqrt(2) ---------------------------
x^2-2

Answer 1

lim [x-->√2] √(3+2x) - √2 - 1 = 0

If you plug √2 in for x directly and calculate, you obtain 0. If x increases or decrease slightly, the function remains defined and correspondingly increases or decreases. Nothing tricky here, just direct substitution.

--charlie

Answer 2

This is only a partial answer. You could try putting into a calculator values of x increasingly close to sqrt2 to see what happens. This will give you an idea of whether there is a limit and if so what it is. Proving it might be harder.

Later edit. I have done it rather quickly, so might be wrong, but I think that the limit is (2 - sqrt2)/4. Proving it is rather long to put here. It involves replacing x by sqrt(2 + a) where a is very small and then using the binomial expansion for (1 + x)^0.5 to a first approximation.

Answer 3

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