anonymous
  • anonymous
limit as x approaches 0 of (1-cos5x)/7x^2 ?
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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JamesJ
  • JamesJ
Do you know this result? \[ \lim_{x \rightarrow 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \] If so, put your equation in that form and you find your answer. Hint: the answer to your problem is not 1/2.
anonymous
  • anonymous
No, how is the form of this 1/2? I know that 1-cosax / ax is 0, but that doesn't seem helpful.
JamesJ
  • JamesJ
One way or another you need to use the derivative of cos x. Do you know l'Hopital's rule?

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JamesJ
  • JamesJ
or the Taylor/McLauren series of cos x?
anonymous
  • anonymous
Yes, but we haven't gotten to it in class yet, so I'm trying to get there without it.
JamesJ
  • JamesJ
how about the taylor series of cos x?
anonymous
  • anonymous
No
JamesJ
  • JamesJ
Well, I'm fairly confident you need some result using the derivative of cos x. For the moment, if you accept the result I gave above, then \[ \frac{1 - \cos(5x)}{7x^2} = \frac{1 - \cos(5x)}{(7/25)(5x)^2} \] \[ = \frac{25}{7} \frac{1-\cos(5x)}{(5x)^2} \] Now the limit of that expression on the right is 1/2, hence the limit of the entire expression is \[ \frac{25}{14}. \]
anonymous
  • anonymous
Yes, I came to the same conclusion using derivatives (and I thank you), but we haven't gotten there in class yet and I hesitate to use them. I had hoped to get there algebraically but I see no option.
anonymous
  • anonymous
By algebraically, I meant manipulation not involving derivatives/l'hopital's rule, that is.
JamesJ
  • JamesJ
Yes, I understand. As I say, I don't think you can escape using the derivative one way or another. The other way besides l'Hopital's rule is the use the power series expansion, the Taylor series: \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... \] You can see from that how the result I wrote down above could be derived.
JamesJ
  • JamesJ
(To derive that series, you need to know the derivatives of cos x.)
anonymous
  • anonymous
As ever, you are a gentlemen and a scholar.

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