Limits

Define the function $g$ by . We will investigate the limit $\displaystyle{\lim_{(x,y)\to(0,0)}g(x,y)}$.

1. What is the behavior of $g$ on the $x$-axis? That is, what is $g(x,0)$ and what is the limit of $g$ as $(x,y)$ approaches $(0,0)$ along the $x$-axis?

2. What is the behavior of $g$ on the $y$-axis? That is, what is $g(0,y)$ and what is the limit of $g$ as $(x,y)$ approaches $(0,0)$ along the $y$-axis?

3. What is the behavior of $g$ on the line $y=mx$? That is, what is $g(x,mx)$ and what is the limit of $g$ as $(x,y)$ appro…

Consider the function $f$ defined by $f(x,y) = \frac{xy}{x^2 + y^2 + 1}.$

1. What is the domain of $f$?

2. Evaluate limit of $f$ at $(0,0)$ along the following paths: $x = 0$, $y = 0$, $y = x$, and $y = x^2$.

3. What do you conjecture is the value of $\lim_{(x,y) \to (0,0)} f(x,y)$?

4. Is $f$ continuous at $(0,0)$? Why or why not?

5. Use appropriate technology to sketch both surface and contour plots of $f$ near $(0,0)$. Write several sentences to say how your plots affirm your findings in (a) - (d).

Consider the function $g$ defined by $g(x,y) = \frac{xy}{x^2 + y^2}.$

1. What is the domain of $g$?

2. Evaluate limit of $g$ at $(0,0)$ along the following paths: $x = 0$, $y = x$, and $y = 2x$.

3. What can you now say about the value of $\lim_{(x,y) \to (0,0)} g(x,y)$?

4. Is $g$ continuous at $(0,0)$? Why or why not?

5. Use appropriate technology to sketch both surface and contour plots of $g$ near $(0,0)$. Write several sentences to say how your plots affirm your findings in (a) - (d).

Consider the function $h$ defined by $h(x,y) = \frac{2x^2y}{x^4 + y^2}.$

1. What is the domain of $h$?

2. Evaluate the limit of $h$ at $(0,0)$ along all linear paths the contain the origin. What does this tell us about $\lim_{(x,y) \to (0,0)} h(x,y)$?

3. Does $\lim_{(x,y) \to (0,0)} h(x,y)$ exist? Verify your answer. Check by using appropriate technology to sketch both surface and contour plots of $h$ near $(0,0)$. Write several sentences to say how your plots affirm your findings about $\lim_{(x,y) \to (0,0)} h(x,y)$.

For each of the following prompts, provide an example of a function of two variables with the desired properties (with justification), or explain why such a function does not exist.

1. A function $p$ that is defined at $(0,0)$, but $\lim_{(x,y) \to (0,0)} p(x,y)$ does not exist.

2. A function $q$ that does not have a limit at $(0,0)$, but that has the same limiting value along any line $y = mx$ as $x \to 0$.

3. A function $r$ that is continuous at $(0,0)$, but $\lim_{(x,y) \to (0,0)} r(x,y)$ does not exist…

Use the properties of continuity to determine the set of points at which each of the following functions is continuous. Justify your answers.

1. The function $f$ defined by $f(x,y) = \frac{x+2y}{x-y}$

2. The function $g$ defined by $g(x,y) = \frac{\sin(x)}{1+e^y}$

3. The function $h$ defined by

4. The function $k$ defined by