Multivariable Calculus Quiz: Limits And Continuity Multivariable

What this quiz covers

This quiz focuses on Limits And Continuity Multivariable, giving you a quick way to practice the rules, question types, and explanations that matter most for Multivariable Calculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $L = 0$; both $F_x(0,0)$ and $F_y(0,0)$ exist and equal 0 since the limit of $F$ exists
2. $L = 0$; $F_x(0,0) = 1$ and $F_y(0,0) = -1$ based on the directional behavior of the numerator (correct answer)
3. No value of $L$ makes $F$ continuous because approaching along $x = y$ gives a different limit than along $x = -y$
4. $L = 0$; $F_x(0,0)$ and $F_y(0,0)$ do not exist because the directional derivatives along the coordinate axes are undefined

Question 2

1. The function is continuous at all points $(x,y)$ where $x$ and $y$ are both rational.
2. The function is continuous at all points $(x,y)$ where at least one of $x$ or $y$ is irrational.
3. The function is continuous nowhere on the plane $\mathbb{R}^2$. (correct answer)
4. The function is continuous only at the origin $(0,0)$.

Question 3

1. The limit does not exist, because paths can be chosen to approach the origin from within the region where $f=1$ and from the region where $f=0$. (correct answer)
2. The limit is 1, because there exist paths to the origin along which the function maintains the value 1 throughout the approach.
3. The limit is 0, because for any linear path $y=mx$ with $m>0$, the function values approach 0 as we get sufficiently close to the origin.
4. The limit is $1/2$, representing the weighted average of the function values in the different regions surrounding the origin.

Question 4

1. The limit is 0, because the limit is 0 along all linear paths of approach to the origin.
2. The limit is $1/2$, which can be found by approaching the origin along the path $y=x^2$.
3. The limit does not exist, because the limit value depends on the path of approach to the origin. (correct answer)
4. The limit cannot be determined without also checking all paths of the form $x=ky$.

Question 5

1. The limit exists and equals 0 because the numerator has higher degree than the denominator along any path
2. The limit does not exist because approaching along $y = mx^2$ gives different values for different values of $m$ (correct answer)
3. The limit exists and equals 0 because polar coordinate substitution shows $|f(r\cos\theta, r\sin\theta)| \leq Cr$ for some constant $C$
4. The limit does not exist because the function is undefined on both coordinate axes near the origin

Question 6

1. $h$ is continuous but not differentiable at $(0,0)$ because the partial derivatives exist but are not continuous there (correct answer)
2. $h$ is continuous and differentiable at $(0,0)$ because all directional derivatives exist and the function approaches 0 uniformly
3. $h$ is neither continuous nor differentiable at $(0,0)$ because the function has different limits along different parabolic paths
4. $h$ is continuous and differentiable at $(0,0)$ because $|h(x,y)| \leq \frac{|x|}{\sqrt{2}}$ which ensures both continuity and differentiability

Question 7

1. $k = 1$; $G$ is differentiable at $(0,0)$ with $\nabla G(0,0) = (0,0)$ because the sine function is smooth
2. $k = 0$; $G$ is not differentiable at $(0,0)$ because the limit defining $\frac{\partial G}{\partial x}(0,0)$ does not exist
3. $k = 1$; $G$ is not differentiable at $(0,0)$ because partial derivatives exist but the function is not differentiable in all directions
4. $k = 1$; $G$ is differentiable at $(0,0)$ but $\nabla G$ is not continuous there because $G_x$ and $G_y$ are not continuous at $(0,0)$ (correct answer)

Question 8

1. $\Lambda$ is continuous at $(0,0,0)$ because $|\Lambda(x,y,z)| \leq \frac{\sqrt{x^2 + y^2 + z^2}}{3}$ by the AM-GM inequality
2. $\Lambda$ is discontinuous at $(0,0,0)$ because along the curve $(t, t, t)$, the function approaches $\frac{1}{3}$ as $t \to 0$
3. $\Lambda$ is continuous at $(0,0,0)$ because in spherical coordinates, $\Lambda(\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi) = \rho f(\phi,\theta)$ where $f$ is bounded (correct answer)
4. $\Lambda$ is discontinuous at $(0,0,0)$ because the function can approach different non-zero values along different straight line paths through the origin

Question 9

1. $\alpha = 1$; $\Xi$ is differentiable at $(0,0)$ with both partial derivatives equal to 0 because of the exponential's smoothness
2. $\alpha = 0$; $\Xi$ is not differentiable at $(0,0)$ because the exponential function grows too rapidly near the origin
3. $\alpha = 1$; $\Xi$ is not differentiable at $(0,0)$ because the partial derivatives exist but are not continuous there (correct answer)
4. $\alpha = 1$; $\Xi$ is differentiable at $(0,0)$ with $\Xi_x(0,0) = \Xi_y(0,0) = 0$, and $\nabla \Xi$ is continuous at $(0,0)$

Question 10

1. $\phi$ is discontinuous at $(0,0)$ because the limit as $(x,y) \to (0,0)$ equals $\sqrt{2}$, not 0
2. $\phi$ is continuous at $(0,0)$ because in polar coordinates, $\phi(r\cos\theta, r\sin\theta) = r(\cos^4\theta + \sin^4\theta) \to 0$ as $r \to 0$ (correct answer)
3. $\phi$ is discontinuous at $(0,0)$ because along the line $x = y$, the function approaches a non-zero constant
4. $\phi$ is continuous at $(0,0)$ because the numerator has degree 4 while the denominator has degree 3, ensuring the limit is 0

Question 11

1. The set of all points $(x,y)$ in $\mathbb{R}^2$ such that $x^2+y^2 > 1$.
2. The set of all points $(x,y)$ in $\mathbb{R}^2$ such that $x^2+y^2 \ge 1$. (correct answer)
3. The set of all points $(x,y)$ in $\mathbb{R}^2$ such that $x^2+y^2 \neq 1$.
4. The set of all points $(x,y)$ in $\mathbb{R}^2$.

Question 12

1. The limit is 0, as can be shown by converting to polar coordinates.
2. The limit is 1, as shown by approaching the origin along the x-axis.
3. The limit does not exist because the limit value depends on the angle of approach $\theta$. (correct answer)
4. The limit does not exist because the function is undefined at the origin.

Question 13

1. The function is continuous at $(0,0)$ because its partial derivatives $f_x(0,0)$ and $f_y(0,0)$ both exist.
2. The function is not continuous at $(0,0)$, and its partial derivatives $f_x(0,0)$ and $f_y(0,0)$ do not exist.
3. The function is continuous at $(0,0)$, but its partial derivatives $f_x(0,0)$ and $f_y(0,0)$ do not exist.
4. The function is not continuous at $(0,0)$, but its partial derivatives $f_x(0,0)$ and $f_y(0,0)$ both exist. (correct answer)

Question 14

1. $S = \{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 > 1 \text{ and } x \neq 0 \}$. (correct answer)
2. $S = \{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 \ge 1 \text{ and } x \neq 0 \}$.
3. $S = \{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 > 1 \}$.
4. $S = \{(x,y) \in \mathbb{R}^2 \mid x > 0 \}$.

Question 15

1. The function $f(x,y)$ must be defined at $(a,b)$ and $f(a,b)=L$.
2. For any real number $m$, the limit $\lim_{x \to a} f(x, b+m(x-a))$ exists and is equal to $L$. (correct answer)
3. If $\lim_{x \to a} f(x, g(x)) = L$ for several different path functions $y=g(x)$, then the overall limit must be $L$.
4. The partial derivatives $f_x(a,b)$ and $f_y(a,b)$ must exist.

Question 16

1. $0$
2. It is impossible to make the function continuous at the origin.
3. $e$
4. $1$ (correct answer)

Question 17

1. The limit does not exist because $\cos(\frac{1}{x^2+y^2})$ oscillates infinitely as $(x,y) \to (0,0)$.
2. The limit is 0, which can be established by applying the Squeeze Theorem with appropriate bounding functions. (correct answer)
3. The limit is 1, because the cosine term approaches $\cos(0)=1$ as $(x,y)$ approaches the origin.
4. The limit depends on the path of approach, equaling 1 along the x-axis and -1 along the y-axis.