Lesson test: Limits and continuity

1.[1p]
What value does $\frac{x y}{x^{2} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the line $y = 3 x$ ? Give it as a decimal.

CorrectNot quite: 0.3

2.[2p]
What value does $\frac{x^{2} - y^{2}}{x^{2} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the line $y = 2 x$ ? Give it as a decimal.

CorrectNot quite: -0.6

3.[2p]
What value does $\frac{x^{2} y}{x^{4} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the parabola $y = 2 x^{2}$ ? Give it as a decimal.

CorrectNot quite: 0.4

4.[2p]
What is true of $\frac{x^{2} y}{x^{4} + y^{2}}$ at the origin?

Its limit is $0$ along every line through the origin, yet the limit does not exist

Correct

Its limit is $0$ , because it is $0$ along every line through the origin

The answer is: Its limit is $0$ along every line through the origin, yet the limit does not exist

Its limit does not exist, because different lines through the origin give different values

The answer is: Its limit is $0$ along every line through the origin, yet the limit does not exist

Its limit is $\frac{1}{2}$ , the value along $y = x^{2}$

The answer is: Its limit is $0$ along every line through the origin, yet the limit does not exist

5.[3p]
For $lim_{(x, y) \to (0, 0)} (2 x - y) = 0$ , what is the largest $δ$ that works when $ε = 0.1$ ? Round to four decimal places.

CorrectNot quite: 0.0447

6.[2p]
Which argument proves that $\frac{x^{2} y}{x^{2} + y^{2}} \to 0$ at the origin?

In polar coordinates it is $r \cos^{2} θ \sin θ$ , whose size is at most $r$ whatever $θ$ is

Correct

Along every line $y = m x$ it tends to $0$

The answer is: In polar coordinates it is $r\cos^2\theta\sin\theta$, whose size is at most $r$ whatever $\theta$ is

Along the $x$ axis and the $y$ axis it is identically $0$

The answer is: In polar coordinates it is $r\cos^2\theta\sin\theta$, whose size is at most $r$ whatever $\theta$ is

Its numerator and denominator both tend to $0$

The answer is: In polar coordinates it is $r\cos^2\theta\sin\theta$, whose size is at most $r$ whatever $\theta$ is

9.[3p]
Which of these functions have a limit at the origin?

Select all that apply

$\frac{x^{2} y^{2}}{x^{2} + y^{2}}$

Correct

$\frac{x^{3} + y^{3}}{x^{2} + y^{2}}$

Correct

$\frac{x y}{x^{2} + y^{2}}$

The answer is: $\dfrac{x^2 y^2}{x^2 + y^2}$, $\dfrac{x^3 + y^3}{x^2 + y^2}$

$\frac{x^{2} - y^{2}}{x^{2} + y^{2}}$

The answer is: $\dfrac{x^2 y^2}{x^2 + y^2}$, $\dfrac{x^3 + y^3}{x^2 + y^2}$

$\frac{x^{2} y}{x^{4} + y^{2}}$

The answer is: $\dfrac{x^2 y^2}{x^2 + y^2}$, $\dfrac{x^3 + y^3}{x^2 + y^2}$

Sign in

Limits and continuity

1.[1p]
What value does $\frac{x y}{x^{2} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the line $y = 3 x$ ? Give it as a decimal.

2.[2p]
What value does $\frac{x^{2} - y^{2}}{x^{2} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the line $y = 2 x$ ? Give it as a decimal.

3.[2p]
What value does $\frac{x^{2} y}{x^{4} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the parabola $y = 2 x^{2}$ ? Give it as a decimal.

4.[2p]
What is true of $\frac{x^{2} y}{x^{4} + y^{2}}$ at the origin?

5.[3p]
For $lim_{(x, y) \to (0, 0)} (2 x - y) = 0$ , what is the largest $δ$ that works when $ε = 0.1$ ? Round to four decimal places.

6.[2p]
Which argument proves that $\frac{x^{2} y}{x^{2} + y^{2}} \to 0$ at the origin?

7.[1p]
Evaluate $lim_{(x, y) \to (1, 2)} \frac{x^{2} y + 3}{x + y}$ , rounded to three decimal places.

8.[2p]
What value at the origin makes $\frac{\ln (1 + x^{2} + y^{2})}{x^{2} + y^{2}}$ continuous there?

9.[3p]
Which of these functions have a limit at the origin?

1.[1p] What value does xyx2+y2 approach as (x,y)→(0,0) along the line y=3x? Give it as a decimal.

2.[2p] What value does x2-y2x2+y2 approach as (x,y)→(0,0) along the line y=2x? Give it as a decimal.

3.[2p] What value does x2yx4+y2 approach as (x,y)→(0,0) along the parabola y=2x2? Give it as a decimal.

4.[2p] What is true of x2yx4+y2 at the origin?

5.[3p] For lim(x,y)→(0,0)(2x-y)=0, what is the largest δ that works when ε=0.1? Round to four decimal places.

6.[2p] Which argument proves that x2yx2+y2→0 at the origin?

7.[1p] Evaluate lim(x,y)→(1,2)x2y+3x+y, rounded to three decimal places.

8.[2p] What value at the origin makes ln(1+x2+y2)x2+y2 continuous there?

9.[3p] Which of these functions have a limit at the origin?

1.[1p]
What value does $\frac{x y}{x^{2} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the line $y = 3 x$ ? Give it as a decimal.

2.[2p]
What value does $\frac{x^{2} - y^{2}}{x^{2} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the line $y = 2 x$ ? Give it as a decimal.

3.[2p]
What value does $\frac{x^{2} y}{x^{4} + y^{2}}$ approach as $(x, y) \to (0, 0)$ along the parabola $y = 2 x^{2}$ ? Give it as a decimal.

4.[2p]
What is true of $\frac{x^{2} y}{x^{4} + y^{2}}$ at the origin?

5.[3p]
For $lim_{(x, y) \to (0, 0)} (2 x - y) = 0$ , what is the largest $δ$ that works when $ε = 0.1$ ? Round to four decimal places.

6.[2p]
Which argument proves that $\frac{x^{2} y}{x^{2} + y^{2}} \to 0$ at the origin?

7.[1p]
Evaluate $lim_{(x, y) \to (1, 2)} \frac{x^{2} y + 3}{x + y}$ , rounded to three decimal places.

8.[2p]
What value at the origin makes $\frac{\ln (1 + x^{2} + y^{2})}{x^{2} + y^{2}}$ continuous there?

9.[3p]
Which of these functions have a limit at the origin?