Limits along different paths on a surface in 3D

Direct answer

Explore 3D surfaces and approach the origin along lines and parabolas to see when a limit exists and when it fails.

Walk toward the origin along different paths (pink). If two paths end at different heights, the limit does not exist.

Simulation

Limits

Surface z = f(x, y) xy / (x² + y²)

Approach angle (°) 45°

Path type Straight line

Drag to rotate, two fingers or right-drag to pan, scroll or pinch to zoom. The toolbar has views, cutaway, measure and “show the maths”.

What's happening? Walk toward the origin along different paths (pink). If two paths end at different heights, the limit does not exist.

Learn more

Where it is in NCERT: Class 11 Mathematics, “Limits and Derivatives” (Limits, Limits of Trigonometric Functions, Algebra of Limits, Derivatives) · ncert.nic.in
Topic notes and formulas: Limits
Printable formula sheet
Chapter-wise JEE Main PYQs (5)