3.1.E: Geometry, Limits, and Continuity (Exercises)

Exercise $\PageIndex{1}$

Plot the graph and a contour plot for each of the following functions. Do your plots over regions large enough to illustrate the behavior of the function.

(a) $f(x, y)=x^{2}+4 y^{2}$

(b) $f(x, y)=x^{2}-y^{2}$

(c) $f(x, y)=4 y^{2}-2 x^{2}$

(d) $h(x, y)=\sin (x) \cos (y)$

(e) $f(x, y)=\sin (x+y)$

(f) $g(x, y)=\sin \left(x^{2}+y^{2}\right)$

(g) $g(x, y)=\sin \left(x^{2}-y^{2}\right)$

(h) $h(x, y)=x e^{-\sqrt{x^{2}+y^{2}}}$

(i) $f(x, y)=\frac{1}{2 \pi} e^{-\frac{1}{2 \pi}\left(x^{2}+y^{2}\right)}$

(j) $f(x, y)=\sin (\pi \sin (x)+y)$

(k) $h(x, y)=\frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}$

(l) $g(x, y)=\log \left(\sqrt{x^{2}+y^{2}}\right)$

Exercise $\PageIndex{2}$

For each of the following, plot the contour surface $f(x,y,z) = c$ for the specified value of $c$

(a) $f(x, y, z)=x^{2}+y^{2}+z^{2}, c=4$

(b) $f(x, y, z)=x^{2}+4 y^{2}+2 z^{2}, c=7$

(c) $f(x, y, z)=x^{2}+y^{2}-z^{2}, c=1$

(d) $f(x, y, z)=x^{2}-y^{2}+z^{2}, c=1$

Exercise $\PageIndex{3}$

Evaluate the following limits.

(a) $\lim _{(x, y) \rightarrow(2,1)}\left(3 x y+x^{2} y+4 y\right)$

(b) $\lim _{(x, y, z) \rightarrow(1,2,1)} \frac{3 x y z}{2 x y^{2}+4 z}$

(c) $\lim _{(x, y) \rightarrow(2,0)} \frac{\cos (3 x y)}{\sqrt{x^{2}+1}}$

(d) $\lim _{(x, y, z) \rightarrow(2,1,3)} y e^{2 x-3 y+z}$

Answer

(a) $\lim _{(x, y) \rightarrow(2,1)}\left(3 x y+x^{2} y+4\right)=14$

(b) $\lim _{(x, y, z) \rightarrow(1,2,1)} \frac{3 x y z}{2 x y^{2}+4 z}=\frac{1}{2}$

(c) $\lim _{(x, y) \rightarrow(2,0)} \frac{\cos (3 x y)}{\sqrt{x^{2}+1}}=\frac{1}{\sqrt{5}}$

(d) $\lim _{(x, y, z) \rightarrow(2,1,3)} y e^{2 x-3 y+z}=e^{4}$

Exercise $\PageIndex{4}$

For each of the following, either find the specified limit or explain why the limit does not exist.

(a) $\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{2}}{x^{2}+y^{2}}$

(b) $\lim _{(x, y) \rightarrow(0,0)} \frac{x}{x+y}$

(c) $\lim _{(x, y) \rightarrow(0,0)} \frac{x}{x+y^{2}}$

(d) $\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}$

(e) $\lim _{(x, y) \rightarrow(0,0)} \frac{1-e^{-\left(x^{2}+y^{2}\right)}}{x^{2}+y^{2}}$

(f) $\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{2}+y^{2}}$

Answer

(a) $\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{2}}{x^{2}+y^{2}}=0$

(c)The limit does not exist: For example, if we let

$$f(x, y)=\frac{x}{x+y^{2}} \nonumber$$

$\alpha(t)=(0, t)$, and $\beta(t)=(t, 0)$, then

$$\lim _{t \rightarrow 0} f(\alpha(t))=0 \nonumber$$

while

$$\lim _{t \rightarrow 0} f(\beta(t))=1. \nonumber$$

(e) $\lim _{(x, y) \rightarrow(0,0)} \frac{1-e^{-\left(x^{2}+y^{2}\right)}}{x^{2}+y^{2}}=1$

Exercise $\PageIndex{5}$

Let $f(x, y)=\frac{x^{2} y}{x^{4}+4 y^{2}}$.

(a) Define $\alpha: \mathbb{R} \rightarrow \mathbb{R}^{2}$ by $\alpha(t)=(t, 0)$. Show that $\lim _{t \rightarrow 0} f(\alpha(t))=0$.

(b) Define $\beta: \mathbb{R} \rightarrow \mathbb{R}^{2}$ by $\beta(t)=(0, t)$. Show that $\lim _{t \rightarrow 0} f(\beta(t))=0$.

(c) Show that for any real number $m$, if we define $\gamma: \mathbb{R} \rightarrow \mathbb{R}^{2}$ by $\gamma(t)=(t, m t)$, then $\lim _{t \rightarrow 0} f(\gamma(t))=0$.

(d) Define $\delta: \mathbb{R} \rightarrow \mathbb{R}^{2}$ by $\delta(t)=\left(t, t^{2}\right)$. Show that $\lim _{t \rightarrow 0} f(\delta(t))=\frac{1}{5}$.

(e) What can you conclude about $\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+4 y^{2}} ?$

(f) Plot the graph of $f$ and explain your results in terms of the graph.

Exercise $\PageIndex{6}$

Discuss the continuity of the function

$$f(x, y)= \begin{cases}\frac{1-e^{-\sqrt{x^{2}+y^{2}}}}{\sqrt{x^{2}+y^{2}}}, & \text { if }(x, y) \neq(0,0), \\ 1, & \text { if }(x, y)=(0,0) .\end{cases} \nonumber$$

Exercise $\PageIndex{7}$

Discuss the continuity of the function

$$g(x, y)= \begin{cases}\frac{x^{2} y^{2}}{x^{4}+y^{4}}, & \text { if }(x, y) \neq(0,0), \\ 1, & \text { if }(x, y)=(0,0) . \end{cases} \nonumber$$

Exercise $\PageIndex{8}$

For each of the following, decide whether the given set is open, closed, neither open nor closed, or both open and closed.

(a) $(3,10) \text { in } \mathbb{R}$

(b) $[-2,5] \text { in } \mathbb{R}$

(c) $\left\{(x, y): x^{2}+y^{2}<4\right\} \text { in } \mathbb{R}^{2}$

(d) $\left\{(x, y): x^{2}+y^{2}>4\right\} \text { in } \mathbb{R}^{2}$

(e) $\left\{(x, y): x^{2}+y^{2} \leq 4\right\} \text { in } \mathbb{R}^{2}$

(f) $\left\{(x, y): x^{2}+y^{2}=4\right\} \text { in } \mathbb{R}^{2}$

(g) $\{(x, y, z):-1<x<1,-2<y<3,2<z<5\} \text { in } \mathbb{R}^{3}$

(h) $\{(x, y):-3<x \leq 4,-2 \leq y<1\} \text { in } \mathbb{R}^{2}$

Answer

(a) Open

(b) Closed

(c) Open

(d) Open

(e) Closed

(f) Closed

(g) Open

(h) Neither open nor closed

Exercise $\PageIndex{9}$

Give an example of a subset of $\mathbb{R}$ which is neither open nor closed.

Exercise $\PageIndex{10}$

Is it possible for a subset of $\mathbb{R}^2$ to be both open and closed? Explain.