$$\lim\limits_{(x, y) \to (a, b)} f(x, y) = f(a, b)$$

$$\lim\limits_{(x, y) \to (a, b)} f(x, y) = f(a, b)$$

Question 1

Let $$f(x, y) = \frac{3x^2$y}{x^2$ + y^2$}. $$What is $$\lim\limits_{(x, y) 	o (0, 0)} f(x, y)$$ along the path $$y = 0$$?

Accepted Answer

$$0$$

Question 2

Which of the following statements is TRUE about the limit $$\lim\limits_{(x, y) 	o (0, 0)} \frac{x^2$ - y^2$}{x^2$ + y^2$}$$?

Accepted Answer

The limit does not exist because the value depends on the path taken.

Question 3

A function $$f of $$two variables is continuous at $$(a, b) if$$:

Accepted Answer

$$\lim\limits_{(x, y) 	o (a, b)} f(x, y) = f(a, b)$$

Question 4

Which of the following is always continuous on $$\mathbb{R}^2$$?

Accepted Answer

Any polynomial function of two variables

Question 5

Given $$f(x, y) = \frac{2xy + 1}{x^2$ + y^2$}$$, which of the following is TRUE?

Accepted Answer

The function is not defined at $$(0, 0)$$

Question 6

Suppose $$f(x, y) = x^2$ + 5x^3$y^2$$ + $$6xy^4 - 7y + 6$. What is $$\lim\limits_{(x, y) 	o (1, 2)} f(x, y)$?

Accepted Answer

$$f(1, 2)$$

Question 7

Which of the following is a necessary condition for $$\lim\limits_{(x, y) 	o (a, b)} f(x, y) = L to $$exist?

Accepted Answer

The value of $$f(x, y)$$ must approach $$L$$ along every possible path to $$(a, b)$$

Question 8

Let $$f(x, y) = \frac{x^2$y}{x^2$ + y^2$}. $$What is $$\lim\limits_{(x, y) 	o (0, 0)} f(x, y)$$ along the path $$y = x$$?

Accepted Answer

$$0$$

Question 9

If $$f(x, y) = \frac{1}{x^2$ + y^2$ - 1}$$, at which of the following points is $$f$$ NOT continuous?

Accepted Answer

$$(1, 0)$$

Question 10

Which of the following best describes a rational function of two variables?

Accepted Answer

A ratio of two polynomials in $$x$$ and $$y$$

Question 11

Suppose $$f(x, y) = x^2$ + y^2$. Which of the following is TRUE?

Accepted Answer

$$f is $$continuous everywhere on $$\mathbb{R}^2$$

Question 12

Let $$f(x, y) = \frac{x^2$ - y^2$}{x^2$ + y^2$}. $$What is the limit as $$(x, y) 	o (0, 0)$$ along the path $$y = x$$?

Accepted Answer

$$0$$

Question 13

Which of the following is a correct statement about the continuity of rational functions?

Accepted Answer

Rational functions are continuous on their domain

Question 14

Let $$f(x, y) = \frac{3x^2$y}{x^2$ + y^2$}. $$What is $$\lim\limits_{(x, y) 	o (0, 0)} f(x, y)$$ along the path $$x = 0$$?

Accepted Answer

$$0$$

Question 15

If $$f(x, y) is $$continuous at $$(a, b)$$, which of the following must be true?

Accepted Answer

$$\lim\limits_{(x, y) 	o (a, b)} f(x, y) = f(a, b)$$

Question 16

Which of the following is a polynomial function of two variables?

Accepted Answer

$$x^2$$ + $$5x^3y^2$ + 6xy^4$ - 7y + 6$

Question 17

Let $$f(x, y) = \frac{x^2$y}{x^2$ + y^2$}. $$Does $$\lim\limits_{(x, y) 	o (0, 0)} f(x, y)$$ exist?

Accepted Answer

Yes, the limit exists and equals $$0$$

Question 18

Which of the following is a correct definition of continuity for a function $$f(x, y) at$$ $$(a, b)$$?

Accepted Answer

$$\lim\limits_{(x, y) 	o (a, b)} f(x, y) = f(a, b)$$

Question 19

Suppose $$f(x, y) = \frac{1}{x^2$ + y^2$ - 1}. Is$$ $$f$$ continuous at $$(0, 0)$$?

Accepted Answer

Yes, because the denominator is not zero at $$(0, 0)$$

Section 4.2 - Limits and Continuity of Two-Variable Functions

Guida di studio - Domande di pratica