- #1
bmxicle
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Homework Statement
This is the solution to an IVP, and the question asks how the function behaves as x Approaches infinity.
[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Carctan%20x%5C%2C%20%2B%5C%2C%20%20x%5Csin%20y%20%5C%2C%2B%5C%2C%20%5Cfrac%7By%5E3%7D%7B6%7D%5C%2C%20%3D%5C%2C%20%5Cfrac%7B2%5Cpi%5E3%5C%2C%2B%5C%2C%203%5Cpi%7D%7B12%7D.gif
Homework Equations
n/a
The Attempt at a Solution
Well i checked the solution on wolfram alpha and it gave me the same solution, so I'm fairly confident my solution is correct and that it wasn't supposed to be a function with an easier limit to compute.
rearranging the equation gives.
[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Csin%20y%20%5C%2C%20%2B%5C%2C%5Cfrac%7By%5E3%7D%7B6x%7D%20%5C%2C%20%3D%20%5C%2C%20%5Cfrac%7B2%5Cpi%5E3%20%2B3%5Cpi%7D%7B12x%7D%5C%2C%20-%5C%2C%20%5Cfrac%7B%5Carctan%20x%7D%7Bx%7D%5C%5C%5Clim_%7Bx%5Cto%5Cinfty%7D%20%5C%2C%20%5Cfrac%7B2%5Cpi%5E3%20%2B3%5Cpi%7D%7B12x%7D%5C%2C-%5C%2C%20%5Cfrac%7B%5Carctan%20x%7D%7Bx%7D%5C%2C%20%3D%20%5C%2C%200%20.gif
The limit of the left hand side of the equation must then be:
[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Clim_%7Bx%5Cto%5Cinfty%7D%20%5Csin%20y%5C%2C%20%2B%5C%2C%5Cfrac%7By%5E3%7D%7B6x%7D%20%5C%2C%20%3D%20%5C%2C%200%20%5C%5C%20%5Cleftrightarrow%20%5C%2C%20y%5C%2C%20%3D%5C%2C%200.gif
So since the right hand side of the solution's limit is zero, for the lefthand side of the solution's limit must also be equal to zero and this occurs iff y = 0.
I'm just not sure if this is a valid approach to computing 'limits' of implicit functions as I don't have Much experience with them yet.
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