What is the Limit of a Function as x Approaches 0?

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The discussion focuses on finding the limit of a function as x approaches 0, specifically using l'Hopital's rule and Taylor series expansions for trigonometric functions. One participant suggests expanding cos(2x) and sin(x) in Taylor series to identify canceling terms, which can simplify the limit calculation. It is recommended to use Taylor series for sine, cosine, and exponential functions as a strategy for solving limit problems. Additionally, factoring out x terms is advised to avoid indeterminate forms. This approach can lead to a clearer path to calculating the limit.
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Homework Statement



http://i45.tinypic.com/20rsnis.jpg

Homework Equations



Compute the lim

The Attempt at a Solution


I tried using l'Hopital's rule but I have to keep finding the derivative and it doesn't yield an answer. I have to use taylor series for the trig functions, but don't know how this will work
 
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Go ahead, try it. Expand cos(2x) and sin(x) in a taylor series. See which terms cancel. If you don't know the expansion of cos and sin, they are easy to look up.
 
http://en.wikipedia.org/wiki/Taylor_series

a good rule of thumb is to turn sin and cosine and even e into their taylor series whenever doing limit problems. then subtract whatever you can. and factor out as many x terms as you need to make the top and bottom non zero. and then calculate the limit.
 

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