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cinarBest ResponseYou've already chosen the best response.0
\[\LARGE \lim_{x \rightarrow a}\frac{x^\sqrt{2}a^\sqrt{2}}{xa}\]
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
I would consider rationalizing the numerator AND denominator first.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Never mind. Factor the numerator.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
@cinar: If that was your answer you are incorrect. I should mention.
 one year ago

cinarBest ResponseYou've already chosen the best response.0
the answer is \[\sqrt{2}a^{\sqrt{2}1}\]if L'Hospital rule is allowed, but without using it, I have no idea..
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
This is the (less often seen) version of the Limit Definition of a Derivative,\[\large f'(x)=\lim_{x \rightarrow a}\frac{f(x)f(a)}{xa}\]So identity f(x), then just take it's derivative.
 one year ago

cinarBest ResponseYou've already chosen the best response.0
I am not looking for derivative of it, it is limit question..
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
This limit represents the LIMIT DEFINITION OF A DERIVATIVE. You are not actually expected to evaluate the limit. You're suppose to recognize that it represents a derivative.
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
Example: Evaluate the limit:\[\large \lim_{h \rightarrow 0}\frac{(x+h)^2x^2}{h}\] You could do the work simplifying it down, Or you can recognize that this is the DERIVATIVE of\[\large f(x)=x^2\]
 one year ago

cinarBest ResponseYou've already chosen the best response.0
\[ f'(x)=\frac{a^{\sqrt2}+\sqrt2 x^{\sqrt21} (xa)x^{\sqrt2}}{(ax)^2}\]
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
\[\large f'(\color{red}{a})=\lim_{x \rightarrow a}\frac{f(x)f(a)}{xa}\]Ah sorry I made a small typo,
 one year ago

cinarBest ResponseYou've already chosen the best response.0
but we cannot simplify it can we?
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
So that's where the a in coming from in the answer i guess :D
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
You just need to recognize that,\[\large f(x)=x^{\sqrt2}\]And that this limit represents,\[\large f'(a)\] So simply take the derivative of f(x), then plug in a! :)
 one year ago

cinarBest ResponseYou've already chosen the best response.0
\[\large \lim_{h \rightarrow 0}\frac{(x+h)^2x^2}{h}=2x ?\]
 one year ago
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