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Limit

What would be the most efficient way to find the limit as x tends to zero of (sinx)^10 / x^10 ? Obviously L'Hopital's works but would take forever, and I'm not sure how I would expand the Taylor series of sin x since its to the power of 10... Thank you!!
Reply 1
Original post by apolaroidofus
What would be the most efficient way to find the limit as x tends to zero of (sinx)^10 / x^10 ? Obviously L'Hopital's works but would take forever, and I'm not sure how I would expand the Taylor series of sin x since its to the power of 10... Thank you!!

(x - x^3/3! + ...)^10
factor out an x to leave (1 - ...)^10 then ...

But the small angle approximation for sin(x) is x, so raising both to the power 10, so its probably the most eficient / write down way.
(edited 8 months ago)
Original post by mqb2766
(x - x^3/3! + ...)^10
factor out an x to leave (1 - ...)^10 then ...
But the small angle approximation for sin(x) is x, so raising both to the power 10, so its probably the most eficient / write down way.

ahhhh thank you
Reply 3
Original post by apolaroidofus
ahhhh thank you

Note you could just do the usual sin(x)/x then raise that result to the power 10.
(edited 8 months ago)
Reply 4
Original post by apolaroidofus
What would be the most efficient way to find the limit as x tends to zero of (sinx)^10 / x^10 ? Obviously L'Hopital's works but would take forever, and I'm not sure how I would expand the Taylor series of sin x since its to the power of 10... Thank you!!

You know the limit of sin x / x. And you know that if functions f, g have limits F, G with F and G not equal to 0 or infinity, then fg has limit FG.

Apply this a few times to find lim (sin x / x)^10.

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