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Mth 277
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** Question Form_labelMessages **
Section 11_2 number 6
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From Section 11.2 homework number 6) Find the limit as (x,y) -> (0,0) of [1 - (sin(x^2 + y^2)/(x^2 + y^2)] or explain why the limit does not exist.
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I would automatically think this limit does not exist because the x^2 + y^2 in the denominator cannot be factored and such. For example, in a case like this Find the limit as (x,y) -> (0,0) of (y - x)/sqrt(x^2 + y^2) where the limit does not exist. I was using Wolfram Alpha to check my work and it says that the limit of the number 6 problem is 0. Basically, I don't understand what you do different between the number 6 problem and the example I just gave where the limit doesn't exist. Why is one 0 and the other DNE?
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In first-year calculus you would almost certainly have proven that the limit as x approaches zero of the expression sin(theta) / theta is 1.
If theta is the angle with the x axis, then on the unit circle the corresponding arc length is also theta. For small theta that arc is almost vertical, hence almost identical with the y coordinate of the unit-circle point corresponding to angle theta.
As theta approaches zero, the arc approaches vertical, so the y coordinate approaches theta.
Thus as theta -> 0, sin(theta) / theta approaches theta / theta = 1.
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For this problem, then, let x^2 + y^2 = theta and the result follows fairly easily.
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