Does Taylor Series accurately represent limits in calculus?

AI Thread Summary
The discussion revolves around the accuracy of Taylor series in representing limits in calculus, specifically as x approaches 0. The user attempted to expand the sine function to the third degree and other functions to the second degree but encountered discrepancies with the expected result from WolframAlpha, which states the limit should be -6/25. The user identified an error in their expansion, noting that the coefficient for the e^3x expansion should be 4.5 instead of 0.5 due to a miscalculation in the factorial term. This highlights the importance of careful expansion and calculation when using Taylor series for limit problems.
ironman
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Homework Statement


[/B]
lim x -> 0
CodeCogsEqn.gif


2. Homework Equations
Taylor series for sin cos e and ln ()

The Attempt at a Solution


I tried expanding the sine to 3-degree, and everything else 2-degree. I ended up with this:

CodeCogsEqn-2.gif

Now the problem is that WolframAlpha says it should be -6/25. Now if only that -2 were +2...
 
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I got the answer! the first 1/2 (expansion from e^3x) is supposed to be 4 1/2 not 1/2, because you get (3x)^2 / 2! not (x)^2/2!
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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