Use L'Hotpital's rule to evaluate?

Use L'Hotpital's rule to evaluate:

lim

x --> 0.... 4x(cos 3x-1) / sin 9x - 9x

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  • 8 years ago
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    It is hard to write down every details but I would suggest you to differentiate top and bottom individually. Remember when you differentiate one time and try to plug the value 0, then you will again encounter the same problem, I mean you will get 0/0 again. So you need to continue with L'Hospital rule. When you do derivative second time, then you will have,

    -36x*cos3x-24sin3x/-81*sin9x

    Now its time to plug the value x=0, then clearly you will have 0/0 again.

    So continue again, this time for sure you will get rid of 0/0 form because , the sine function on the denominator will changed to cosine function and cos0 is equal to 1. And from the top, you ll have some crap (coming from the derivative of -36x*cos3x, which will eventually vanish when you plug x=0) and -24*3cos3x. On plugging x=0 that will give you -72. So you have now -72/-81*9, which is finally your answer. Do not forgot to cancel negative sign and 9 goes 8 times to 72, then you ultimately get 8/81. Hope that helps.

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