Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §13.2 Partial Derivatives, #48 · Multivariable Calculus · Partial derivatives

In (d) find the limit at … along the line

This problem is about limits in two dimensions.

Write down the limit $L$ as $(x, y) \to (a, b)$ . At which points $(a, b)$ does $f (x, y)$ have no limit?
(a) $f (x, y) = x^{2} + y^{2}$ (b) $f (x, y) = x / y$
(c) $f (x, y) = 1/ (x + y)$ (d) $f (x, y) = x y / (x^{2} + y^{2})$

In (d) find the limit at $(0, 0)$ along the line $y = m x$ . The limit changes with $m$ , so $L$ does not exist at $(0, 0)$ . Same for $x / y$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §13.2 Partial Derivatives, #48

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