Can Xy X2 Y2 Be Made Continuous by Suitably Defining It at 0 0
Problem 1
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ and suppose that $\lim _{(x, y) \rightarrow(1,3)} f(x, y)=5 .$ What can you say about the value $f(1,3) ?$.
Nick Johnson
Numerade Educator
Problem 2
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is continuous and suppose that $\lim _{(x, y) \rightarrow(1,3)} f(x, y)=5 .$ What can you say about the value $f(1,3) ?$
Nick Johnson
Numerade Educator
Problem 3
Compute the limits:
(a) $\operatorname{limit}_{(x, y) \rightarrow(0,1)} x^{3} y$
(b) $\operatorname{limit}_{x \rightarrow 0} \frac{\cos x-1}{x^{2}}$
(c) limit $\frac{e^{h}-1}{h}$
Nick Johnson
Numerade Educator
Problem 4
Compute the following limits:$$\text { (a) } \begin{array}{c}\text { linitit } \\(x, y)=(0,1)\end{array} e^{x} y$$,$$\text { (b) } \operatorname{limit}_{x \rightarrow 0} \frac{\sin ^{2} x}{x}$$,$$\text { (c) limit } \frac{\sin ^{2} x}{x-9}$$
Nick Johnson
Numerade Educator
Problem 5
Compute the following limits:
$$\text { (a) } \operatorname{limit}_{x \rightarrow 3}\left(x^{2}-3 x+5\right)$$,(b) limit sin $x$ $x \rightarrow(x)$ $$\text { (c) limit } \frac{(x+h)^{2}-x^{2}}{h}$$
Nick Johnson
Numerade Educator
Problem 6
Let $$f(x, y)=\left\{\begin{array}{cl}\frac{x y^{3}}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\
0 & \text { if }(x, y)=(0,0)\end{array}\right.$$,(a) Compute the limit as $(x, y) \rightarrow(0,0)$ of $f$ along the path $x=0$
(b) Compute the limit as $(x, y) \rightarrow(0,0)$ of $f$ along the path $x=y^{3}$
(c) Show that $f$ is not continuous at (0,0)
Nick Johnson
Numerade Educator
Problem 7
Let $f(x, y, z)=\frac{e^{x+y}}{1+z^{2}},$ Compute
$\lim _{h \rightarrow 0} \frac{f(1,2+h, 3)-f(1,2,3)}{h}$
Nick Johnson
Numerade Educator
Problem 8
Compute the following limits if they exist:
$$\text { (a) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{(x+y)^{2}-(x-y)^{2}}{x y}$$ $$\text { (b) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{\sin x y}{y}$$ $$\text { (c) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{x^{3}-y^{3}}{x^{2}+y^{2}}$$
Nick Johnson
Numerade Educator
Problem 9
Compute the following limits if they exist:$$\text { (a) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{e^{x y}-1}{y}$$ $$\text { (b) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{\cos (x y)-1}{x^{2} y^{2}}$$ $$\text { (c) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{x y}{x^{2}+y^{2}+2}$$
Nick Johnson
Numerade Educator
Problem 10
Compute the following limits if they exist.$$\text { (a) limit }_{(x, y) \rightarrow(0,0)} \frac{e^{x y}}{x+1}$$ $$\text { (b) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{\cos x-1-\left(x^{2} / 2\right)}{x^{4}+y^{4}}$$ $$\text { (c) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{x^{2}+y^{2}}$$
Nick Johnson
Numerade Educator
Problem 11
Compute the following limits if they exist: $$\text { (a) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{\sin x y}{x y}$$ $$\text { (b) } \operatorname{limit}_{(x, y, z) \rightarrow(0,0,0)} \frac{\sin (x y z)}{x y z}$$ $$\begin{aligned}
&\text { (c) } \operatorname{limit}_{(x, x, z) \rightarrow(0,0,0)} f(x, y, z), \text { where } f(x, y, z)=\\&\left(x^{2}+3 y^{2}\right) /(x+1)\end{aligned}$$
Nick Johnson
Numerade Educator
Problem 12
Compute the following limits if they exist.$$\text { (a) } \operatorname{limit}_{x \rightarrow 0} \frac{\sin 2 x-2 x}{x^{3}}$$ $$\text { (b) } \operatorname{limit}_{(x, y) \rightarrow(0,0)} \frac{\sin 2 x-2 x+y}{x^{3}+y}$$ $$\text { (c) } \operatorname{limit}_{(x, y, z) \rightarrow(0,0,0)} \frac{2 x^{2} y \cos z}{x^{2}+y^{2}}$$
Nick Johnson
Numerade Educator
Problem 13
Compute limits $\rightarrow \mathbf{x}_{0} f(\mathbf{x}),$ if it exists, for the following cases: $$\text { (a) } f: \mathbb{R} \rightarrow \mathbb{R}_{1} x \mapsto|x|_{1} x_{0}=1$$ $$\text { (b) } f: \mathbb{R}^{n} \rightarrow \mathbb{R}, \mathbf{x} \mapsto\|\mathbf{x}\|, \text { arbitrary } \mathbf{x}_{0}$$ $$\text { (c) } f: \mathbb{R} \rightarrow \mathbb{R}^{2}, x \mapsto\left(x^{2}, e^{x}\right), x_{0}=1$$ $$\begin{aligned}
&\text { (d) } f: \mathbb{R}^{2} \backslash\{(0,0)\} \rightarrow \mathbb{R}^{2},(x, y) \mapsto(\sin (x-y)\\
&\left.e^{x(y+1)}-x-1\right) /[(x, y)] |, \mathbf{x}_{0}=(0,0)
\end{aligned}$$
Problem 14
Let $f(x, y, z)=\frac{1}{x^{2}+y^{2}+z^{2}-1} .$ Describe geometrically the set in $\mathrm{R}^{3}$ where $f$ fails to be continuous.
Nick Johnson
Numerade Educator
Problem 15
Where is the function $f(x, y)=\frac{1}{x^{2}+y^{2}}$ continuous?
Nick Johnson
Numerade Educator
Problem 16
Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$
(a) Considering $A: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ as a linear map, explicitly write the component functions of $A$
(b) Show that $A$ is continuous on all of $\mathbb{R}^{2}$.
Nick Johnson
Numerade Educator
Problem 17
Find $\lim _{(x, y) \rightarrow(0,0)}\left(3 x^{2}+3 y^{2}\right) \log \left(x^{2}+y^{2}\right) .$ (Hint. Use polar coordinates.)
Nick Johnson
Numerade Educator
Problem 18
Show that the subsets of the plane.$$A=\{(x, y) |-1<x<1,-1<y<1\}$$
Nick Johnson
Numerade Educator
Problem 19
Show that the subsets of the plane.$$B=\{(x, y) | y>0\}$$
Nick Johnson
Numerade Educator
Problem 20
Show that the subsets of the plane.$$C=\left\{(x, y) | 2<x^{2}+y^{2}<4\right\}$$
Nick Johnson
Numerade Educator
Problem 21
Show that the subsets of the plane.$$D=\{(x, y) | x \neq 0 \text { and } y \neq 0\}$$
Problem 22
Let $A \subset \mathbb{R}^{2}$ be the open unit disk $D_{1}(0,0)$ with the point $\mathbf{x}_{0}=(1,0)$ added and let $f: A \rightarrow \mathbb{R}, \mathbf{x} \mapsto f(\mathbf{x})$ be the constant function $f(\mathbf{x})=1 .$ Show that $\operatorname{limit}_{\mathbf{x} \rightarrow \mathbf{x}_{0}} f(\mathbf{x})=1$
Nick Johnson
Numerade Educator
Problem 23
If $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ are continuous, show that the functions $$f^{2} g: \mathbb{R}^{n} \rightarrow \mathbb{R},\mathbf{x} \mapsto[f(\mathbf{x})]^{2} g(\mathbf{x})$$ and $$f^{2}+g: \mathbb{R}^{n} \rightarrow \mathbb{R}, x \mapsto[f(x)]^{2}+g(x)$$ are continuous.
Nick Johnson
Numerade Educator
Problem 24
(a) Show that $f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto(1-x)^{8}+\cos \left(1+x^{3}\right)$ is continuous.
(b) Show that the map $f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto x^{2} e^{x} /(2-\sin x)$ is continuous.
Nick Johnson
Numerade Educator
Problem 25
(a) Can $[\sin (x+y)] /(x+y)$ be made continuous by suitably defining it at (0,0)$?$
(b) $\operatorname{Can} x y /\left(x^{2}+y^{2}\right)$ be made continuous by suitably defining it at (0,0)$?$
(c) Prove that $f: \mathbb{R}^{2} \rightarrow \mathbb{R},(x, y) \mapsto y e^{x}+\sin x+(x y)^{4}$ is
continuous.
Nick Johnson
Numerade Educator
Problem 26
Using either $\varepsilon$ 's and 8 's or spherical coordinates, show that $$\operatorname{limit}_{(x, y, z) \rightarrow(0,0,0)} \frac{x y z}{x^{2}+y^{2}+z^{2}}=0$$
Nick Johnson
Numerade Educator
Problem 27
Use the $\varepsilon-\delta$ formulation of limits to prove that $x^{2} \rightarrow 4$ as $x \rightarrow 2 .$ Give another proof using Theorem 3.
Nick Johnson
Numerade Educator
Problem 28
(a) Prove that for $\mathbf{x} \in \mathbb{R}^{n}$ and $s<t, D_{s}(\mathbf{x}) \subset D_{t}(\mathbf{x})$.
(b) Prove that if $U$ and $V$ are neighborhoods of $\mathbf{x} \in \mathbb{R}^{n}$ then so are $U \cap V$ and $U \cup V$.
(c) Prove that the boundary points of an open interval $(a, b) \subset \mathbb{R}$ are the points $a$ and $b$.
Problem 29
Suppose $x$ and $y$ are in $R^{n}$ and $x \neq y .$ Show that there is a continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ with $f(\mathbf{x})=1$ $f(y)=0,$ and $0 \leq f(z) \leq 1$ for every $z$ in $\mathbb{R}^{n}$.
Nick Johnson
Numerade Educator
Problem 30
Let $f: A \subset \mathbb{R}^{n} \rightarrow \mathbb{R}$ be given and let $\mathbf{x}_{0}$ be a boundary point of $A$. We say that $\operatorname{limit}_{x \rightarrow x_{0}} f(x)=\infty$ if for every $N>0$ there is a $\delta>0$ such that $0<\|\mathbf{x}-\mathbf{x} 0\|<\delta$ and $\mathbf{x} \in A$ implies $f(\mathbf{x})>N$
(a) Prove that limit $_{x \rightarrow 1}(x-1)^{-2}=\infty$
(b) Prove that limit $_{x \rightarrow 0} 1 /|x|=\infty .$ Is it true that limit $_{x \rightarrow 0} 1 / x=\infty ?$
(c) Prove that $\operatorname{limit}_{(x, y) \rightarrow(0,0)} 1 /\left(x^{2}+y^{2}\right)=\infty$
Nick Derr
Numerade Educator
Problem 31
Let $b \in \mathbb{R}$ and $f: \mathbb{R} \backslash[b] \rightarrow \mathbb{R}$ be a function. We write limit $_{x \rightarrow b-} f(x)=L$ and say that $L$ is the left-hand limit of $f$ at $b$ if for every $\varepsilon>0$ there is a $\delta>0$ such that $x b$ and $0<|x-b|<8$ implies $|f(x)-L|<\varepsilon$.(a) Formulate a definition of right-hand limit, or $\operatorname{limit}_{x \rightarrow b+} f(x)$
(b) Find limit $_{x \rightarrow 0-} 1 /\left(1+e^{1 / x}\right)$ and $\operatorname{limit}_{x \rightarrow 0+} 1 /\left(1+e^{1 / x}\right)$
(c) Sketch the graph of $1 /\left(1+e^{1 / x}\right)$.
Problem 32
Show that $f$ is continuous at $\mathbf{x}_{0}$ if and only if
$$\operatorname{limit}_{\mathbf{x} \rightarrow \mathbf{x}_{0}} \mathbb{|} f(\mathbf{x})-f\left(\mathbf{x}_{0}\right) \|=0$$
Problem 33
Let $f: A \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ satisfy.If $(x)-f(y) I \leq K\|x-y\|^{\alpha}$ for all $x$ and $y$ in $A$ for positive constants $K$ and $\alpha$. Show that $f$ is continuous. (Such functions are called Holder-continuous or, if $\alpha=1, \text { Lipschitz-continuous. })$
Problem 34
Show that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is continuous at all points if and only if the inverse image of every open set is open.
Problem 35
(a) Find a specific number $\delta>0$ such that if $|a|<\delta$, then $\left|a^{3}+3 a^{2}+a\right|<1 / 100$
(b) Find a specific number $8>0$ such that if $x^{2}+y^{2}<8^{2},$ then $$\left|x^{2}+y^{2}+3 x y+180 x y^{5}\right|<1 / 10,000$$
Source: https://www.numerade.com/books/chapter/differentiation-17/?section=16057
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