Answer to Question #299767 in Calculus for Rama

Question #299767

Check the limit of the function f(x,y) = 3x^2y/(x^2 + y^2) at origin exist or not

1
Expert's answer
2022-02-22T08:22:36-0500

Let us use the polar coordinates {x=rcosθy=rsinθ\begin{cases} x=r\cos \theta \\ y=r\sin\theta \end{cases} , then the limit (x,y)(0,0)(x,y)\to (0,0) corresponds to the limit r0r\to 0. We have

f(r,θ)=3r3cos2θsinθr2=3rcos2θsinθf(r,\theta)=\frac{3r^3\cos^2\theta \sin \theta}{r^2}=3r\cos^2\theta \sin \theta

We see that the limit limr0f(r,θ)=0\lim_{r\to 0} f(r,\theta) =0 exists and independent of θ\theta (as 3cos2θsinθ3\cos^2\theta \sin \theta is a bounded quantity, so limr0r(bounded quantity)=0\lim_{r\to 0}r\cdot (\text{bounded quantity})=0)


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