Multivariable Calculus · free preview

Limits of Functions of Two Variables

The definition of a limit for functions of several variables, the path test that disproves them, and the bounding arguments that prove them.

1. A two-variable limit must survive every approach

A two-variable limit must survive every approach

2. One bad path vetoes the whole limit

One bad path vetoes the whole limit

3. Polar coordinates can turn a veto test into a proof

Polar coordinates can turn a veto test into a proof

4. Limits follow the same arithmetic as before

Limits follow the same arithmetic as before

5. A limit killed by two paths

What is $\lim_{(x,y) \to (0,0)} \dfrac{x^2 - y^2}{x^2 + y^2}$ ?

6. Testing with the diagonal and beyond

A student tests $f(x, y) = \dfrac{xy}{x^2 + y^2}$ near $(0, 0)$ along the two axes, finds the value $0$ both times, and concludes the limit is $0$ . What is the correct assessment?

7. A polar bound in action

Using polar coordinates, what bound controls $\left|\dfrac{x^3 - y^3}{x^2 + y^2}\right|$ near the origin, and what limit follows?

8. A product approaching through continuous pieces

Compute $\lim_{(x,y) \to (2,3)} \dfrac{x^2 y - 1}{x + y}$ and give the result as a fraction like 11/5.

9. A parabola's vote

Compute $\lim_{(x,y) \to (0,0)} \dfrac{x^2 + 2y^2}{x^2 + y^2}$ along the parabola $y = x^2$ and give the value it produces, as a fraction like 3/2 or a whole number.

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