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cinar

  • 2 years ago

limit question. No L'Hospital rule..

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  1. cinar
    • 2 years ago
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    \[\LARGE \lim_{x \rightarrow a}\frac{x^\sqrt{2}-a^\sqrt{2}}{x-a}\]

  2. Dido525
    • 2 years ago
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    I would consider rationalizing the numerator AND denominator first.

  3. cinar
    • 2 years ago
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    sqrt2 power of x and a

  4. Dido525
    • 2 years ago
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    Never mind. Factor the numerator.

  5. Dido525
    • 2 years ago
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    @cinar .

  6. Dido525
    • 2 years ago
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    you there?

  7. Dido525
    • 2 years ago
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    @cinar: If that was your answer you are incorrect. I should mention.

  8. cinar
    • 2 years ago
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    |dw:1355956723907:dw|

  9. cinar
    • 2 years ago
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    the answer is \[\sqrt{2}a^{\sqrt{2}-1}\]if L'Hospital rule is allowed, but without using it, I have no idea..

  10. zepdrix
    • 2 years ago
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    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.

  11. cinar
    • 2 years ago
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    I am not looking for derivative of it, it is limit question..

  12. zepdrix
    • 2 years ago
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    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.

  13. zepdrix
    • 2 years ago
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    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\[\large f(x)=x^2\]

  14. cinar
    • 2 years ago
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    \[ f'(x)=\frac{a^{\sqrt2}+\sqrt2 x^{\sqrt2-1} (x-a)-x^{\sqrt2}}{(a-x)^2}\]

  15. zepdrix
    • 2 years ago
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    ...?

  16. cinar
    • 2 years ago
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    \[f(x)=x^{\sqrt2} ?\]

  17. zepdrix
    • 2 years ago
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    Yes, good.

  18. cinar
    • 2 years ago
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    I see now..

  19. cinar
    • 2 years ago
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    thanks..

  20. zepdrix
    • 2 years ago
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    \[\large f'(\color{red}{a})=\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}\]Ah sorry I made a small typo,

  21. cinar
    • 2 years ago
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    but we cannot simplify it can we?

  22. zepdrix
    • 2 years ago
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    So that's where the a in coming from in the answer i guess :D

  23. zepdrix
    • 2 years ago
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    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! :)

  24. cinar
    • 2 years ago
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    cool..

  25. cinar
    • 2 years ago
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    \[\large \lim_{h \rightarrow 0}\frac{(x+h)^2-x^2}{h}=2x ?\]

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