GATE 2025 MA – Question 26

Calculus · Multivariable Differentiation and Optimization · 1 mark · Multiple select

For (x,y)≠(0,0), $f=(x^2-y^2)/(x^2+y^2)+x\sin[1/(x^2+y^2)]$. S1: lim_(x→0) lim_(y→0) f exists; S2: lim_(y→0) lim_(x→0) f exists; S3: joint limit at origin exists. Which statements hold?

S2,S3 true; S1 false
S1,S2 true; S3 false
S1,S3 true; S2 false
All three true

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Show answer and explanation

Correct answer: (B) S1,S2 true; S3 false

Explanation

First y→0 gives 1+x sin(1/x²), whose x→0 limit is 1. First x→0 gives −1, also with y→0 limit −1. Thus both iterated limits exist but differ. The joint limit cannot exist, as also seen along the two axes. **B**.

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