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

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

Question 1

Let $f(x, y) = \frac{3x^2y}{x^2 + y^2}$. 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 $f(x, y) = x^2 + 5x^3y^2 + 6xy^4 - 7y + 6$. 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 $f(x, y) = \frac{x^2y}{x^2 + y^2}$. 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 $f(x, y) = \frac{3x^2y}{x^2 + y^2}$. 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 $f(x, y) = \frac{x^2y}{x^2 + y^2}$. 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

Study Guide - Practice Questions