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Babynini

  • one year ago

More limits..

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  1. Babynini
    • one year ago
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  2. Babynini
    • one year ago
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    I'm trying to rationalize but I keep going off somewhere.

  3. Empty
    • one year ago
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    Yeah that would be my first choice too, to multiply by this form of 1 and see what happens: \[\large \frac{\sqrt{x^2+9}+5}{\sqrt{x^2+9}+5}\]

  4. anonymous
    • one year ago
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    \[\lim_{x \rightarrow -4}\frac{ \sqrt{x^2+9}-5 }{ x+4 }\]

  5. Babynini
    • one year ago
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    @Empty the problem is i'm not quite sure how to multiply those..heh

  6. Jhannybean
    • one year ago
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    hint: First define what type of form it falls under

  7. Jhannybean
    • one year ago
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    Just judging by the left and right hand limit type, and the form, you can apply LH rule to this.

  8. Empty
    • one year ago
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    In a lot of ways I really look at calculus as 'mastering algebra' Let's just look at the numerator or denominator one at at time, first the numerator: \[(\sqrt{x^2+9}-5)(\sqrt{x^2+9}+5)\] What do we do? Well, it turns out we distribute, just like we would here even though it looks more complicated, try to fit it to this pattern if it helps you keep things straight: \[(a-b)(a+b)=a*a+a*b-b*a-b*b = a^2-b^2\] (notice the middle terms cancel out, which is super handy to save time) Give it a shot, the work you put in now will make a difference later in the semester and during tests.

  9. anonymous
    • one year ago
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    \[\left( \sqrt{x^2+9}-5 \right) \times \frac{ \left( \sqrt{x^2+9}+5 \right) }{ \left( \sqrt{x^2+9}+5 \right) }=\frac{ \left( \sqrt{x^2+9} \right)^2-5^2}{\sqrt{x^2+9}+5}\]

  10. Babynini
    • one year ago
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    ah k @Empty let me try hahaa

  11. Jhannybean
    • one year ago
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    So you want to simply this function to a form where you can input your limit value to break free from the \(\dfrac{0}{0}\) form. Take the derivative of the numerator an the denominator. \[\lim_{x\rightarrow -4} \frac{\frac{d}{dx}(\sqrt{x^2+9}-5)}{\frac{d}{dx}(x+4)}\]\[\lim_{x\rightarrow -4}\frac{\frac{1}{2}(x^2+9)^{-1/2} \cdot 2x}{1}\]\[\lim_{x\rightarrow -4} \frac{\dfrac{x}{(x^2+9)^{1/2}}}{1}\]

  12. Babynini
    • one year ago
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    so for the numerator, does it equal: x^2 + 9 - 25?

  13. Jhannybean
    • one year ago
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    \[\lim_{x\rightarrow -4} \frac{\dfrac{x}{(x^2+9)^{1/2}}}{1} =\frac{-4}{((-4)^2+9)^{1/2}} = \color{red}{-\frac{4}{5}} \]

  14. Babynini
    • one year ago
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    I'm having trouble with the denominator :|

  15. Babynini
    • one year ago
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    @Jhannybean I'm not really sure what all you did because I have no idea how to do derivatives or anything.. so I need to do it algebraically o.o

  16. Jhannybean
    • one year ago
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    Yeah i just realized that, sorry about that

  17. Empty
    • one year ago
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    Yes you're correct with the numerator, however there are two simplifications you can do! combine the 9 and -25 and then you can factor this as the difference of two squares. Try that out, and I'll help you worry about the bottom since you don't want to distribute the bottom actually!

  18. Babynini
    • one year ago
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    Okay! the top then is (x-4)(x+4)

  19. Empty
    • one year ago
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    Awesome. Notice anything about the denominator now? :D

  20. Babynini
    • one year ago
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    (Sorry, went to have lunch)

  21. Babynini
    • one year ago
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    It has an x+4 in it!

  22. Babynini
    • one year ago
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    \[\frac{ x-4 }{ \sqrt{x^2+9} }\]

  23. Babynini
    • one year ago
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    (with a +5 in the denominator, sorry)

  24. Empty
    • one year ago
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    Awesome, now try taking the limit. :P

  25. Babynini
    • one year ago
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    em it says error o.o

  26. Babynini
    • one year ago
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    (-4-4)/ [sq{-4^2+9)}+5]

  27. Babynini
    • one year ago
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    @Empty

  28. zepdrix
    • one year ago
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    do it :U

  29. Babynini
    • one year ago
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    It's not working! because it ends up being a negative under the sq root..

  30. zepdrix
    • one year ago
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    \(\large\rm -4^2\ne(-4)^2\) careful how you square.

  31. Babynini
    • one year ago
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    oh. -8/10

  32. zepdrix
    • one year ago
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    yay c:

  33. IrishBoy123
    • one year ago
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    drum roll!!!

  34. Babynini
    • one year ago
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    hahah

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