11.2#6

Hello professor Taylor, 

I am sending you the picture of my work for problem 6 of 11.2 webwork.  I have simpliefed the expression after using polar coordinates but I don't know what next after there!  Need your help!

Thanks, 

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What you do after is think this way:  

"Well, now I am taking the limit of a function that is the product of two factors, r and some junk that depends on θ.  The r factor is interesting because it just measures how far (x,y) is from (0,0), and since (x,y)-->(0,0) that means that the limit of r is just zero.  At the same time, the junk involving θ, although it doesn't really have a limit, it is staying bounded above and below: since -1 ≤ cos(θ) ≤ 1 and -1 ≤ sin(θ) ≤ 1 we get   

-10 ≤ cos^3(θ) + 9 sin^3(θ) ≤ 10.

This means that  -10 r ≤ r(cos^3(θ) + 9 sin^3(θ) )≤ 10 r.   Since both 10 r and -10 r have zero as a limit, the limit of my function must be zero too!"