IGNOU BMTC 102 2026 - Using polar coordinates, show that . Also, find the two rep

Question

Using polar coordinates, show that $\displaystyle \lim_{\substack{x \to 0 \\ y \to 0}} \frac{x^3 - y^3}{x^2 + y^2} = 0$ . Also, find the two repeated limits.

09 Jan 2026

Answer :

Word Count : 228

Let ( f(x, y) = \frac{x^3 - y^3}{x^2 + y^2} ). To examine the limit as ((x, y) \to (0, 0)) using polar coordinates, set [ x = r \cos\theta, \quad y = r \sin\theta, \quad r \ge 0, \quad 0 \le \theta < 2\pi. ] Then [ x^2 + y^2 = r^2 (\cos^2\theta + \sin^2\theta) = r^2, ] [ x^3 - y^3 = r^3 (\cos^3\theta - \sin^3\theta). ] Thus the function becomes [ f(x, y) = \frac{x^3 - y^3}{x^2 + y^2} = \frac{r^3 (\cos^3\theta - \sin^3\theta)}{r^2}

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