Vault Of Euclid

Epsilon-Delta Proof for a Multivariable Limit

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Let $f(x, y) = \frac{5x^2 y^2}{x^2 + y^2}$. Find $\lim_{(x,y) \to (0,0)} f(x, y)$.

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What you need first

The epsilon-delta definition of a limit for functions of two variables
Testing limits along lines $y=mx$
Basic algebraic inequalities

What this problem tests

Tests the ability to bound a multivariable expression in terms of the distance to the origin to satisfy the formal definition of a limit.

This is a standard rigorous limit proof found in multivariable calculus courses.

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