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Part CCSIR NET December 2025approaching-along-each-axis-gives-a-and-b-respectively-distinctness-of-a-and-b-is-exactly-what-breaks-the-limit

Approaching along each axis gives a and b respectively distinctness of a and b is exactly what breaks the limit

Let a,b be distinct positive real numbers. Consider the function f:R2→Rf:\mathbb{R}^{2}\to\mathbb{R} given by f(x,y)=(ax+by)2/()^{2}/(ax2+^{2}+by2)^{2}) if (x,y)≠(0,0), and 0 if (x,y)=(0,0). Which of the following statements are necessarily true?

  1. A.lim(x,y)→(0,0) f(x,y) does not exist.✓
  2. B.The partial derivatives of f at (0,0) do not exist.✓
  3. C.lim(x→0) f(x,0) = lim(y→0) f(0,y).
  4. D.f is differentiable at (0,0).

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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50 are analysed free — try those first.

Related counterexample: If all partial derivatives exist at a point then f is continuous there

More on this topic

The chapter behind this: Differentiability in several variables — free to read

From Functions of Several Variables › Partial derivatives, differentiability, chain rule

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