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.