3. Partial Derivatives
Limits And Continuity — Quiz
Test your understanding of limits and continuity with 5 practice questions.
Practice Questions
Question 1
Evaluate the limit: $\lim_{(x,y) \to (0,0)} \frac{x^2 y}{x^2 + y^2}$.
Question 2
Determine whether the following function is continuous at $(0,0)$:
$$f(x,y) = \begin{cases} \frac{x^2 - y^2}{x^2 + y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases}$$
$$f(x,y) = \begin{cases} \frac{x^2 - y^2}{x^2 + y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{cases}$$
Question 3
Find the limit: $\lim_{(x,y) \to (0,0)} \frac{x^4 + y^4}{x^2 + y^2}$.
Question 4
Consider the function $f(x,y) = \frac{x^2 y}{x^2 + y^2}$. For which of the following paths does the limit as $(x,y) \to (0,0)$ equal $\frac{1}{2}$?
Question 5
Which of the following statements about the continuity of the function $f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}$ is true?