anonymous
  • anonymous
limit question. No L'Hospital rule..
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
\[\LARGE \lim_{x \rightarrow a}\frac{x^\sqrt{2}-a^\sqrt{2}}{x-a}\]
anonymous
  • anonymous
I would consider rationalizing the numerator AND denominator first.
anonymous
  • anonymous
sqrt2 power of x and a

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anonymous
  • anonymous
Never mind. Factor the numerator.
anonymous
  • anonymous
@cinar .
anonymous
  • anonymous
you there?
anonymous
  • anonymous
@cinar: If that was your answer you are incorrect. I should mention.
anonymous
  • anonymous
|dw:1355956723907:dw|
anonymous
  • anonymous
the answer is \[\sqrt{2}a^{\sqrt{2}-1}\]if L'Hospital rule is allowed, but without using it, I have no idea..
zepdrix
  • zepdrix
This is the (less often seen) version of the Limit Definition of a Derivative,\[\large f'(x)=\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}\]So identity f(x), then just take it's derivative.
anonymous
  • anonymous
I am not looking for derivative of it, it is limit question..
zepdrix
  • zepdrix
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.
zepdrix
  • zepdrix
Example: Evaluate the limit:\[\large \lim_{h \rightarrow 0}\frac{(x+h)^2-x^2}{h}\] You could do the work simplifying it down, Or you can recognize that this is the DERIVATIVE of\[\large f(x)=x^2\]
anonymous
  • anonymous
\[ f'(x)=\frac{a^{\sqrt2}+\sqrt2 x^{\sqrt2-1} (x-a)-x^{\sqrt2}}{(a-x)^2}\]
zepdrix
  • zepdrix
...?
anonymous
  • anonymous
\[f(x)=x^{\sqrt2} ?\]
zepdrix
  • zepdrix
Yes, good.
anonymous
  • anonymous
I see now..
anonymous
  • anonymous
thanks..
zepdrix
  • zepdrix
\[\large f'(\color{red}{a})=\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}\]Ah sorry I made a small typo,
anonymous
  • anonymous
but we cannot simplify it can we?
zepdrix
  • zepdrix
So that's where the a in coming from in the answer i guess :D
zepdrix
  • zepdrix
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! :)
anonymous
  • anonymous
cool..
anonymous
  • anonymous
\[\large \lim_{h \rightarrow 0}\frac{(x+h)^2-x^2}{h}=2x ?\]

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