Question 13.48

Let \(f\colon\mathbb{R}^2 \to \mathbb{R}\) and suppose that \(\displaystyle\lim_{(x,y)\rightarrow(1,3)} f(x, y) =5\). What can you say about the value \(f(1,3)\)?

Question 13.49

Let \(f\colon\mathbb{R}^2 \to \mathbb{R}\) is continuous and suppose that \(\lim_{(x,y)\rightarrow(1,3)} f(x, y) =5\). What can you say about the value \(f(1,3)\)?

Question 13.50

Compute the limits:

• (a) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,1)} x^3y\)
• (b) \(\displaystyle\mathop{\rm limit}_{x\rightarrow 0} \frac{\cos x-1}{x^2}\)
• (c) \(\displaystyle\mathop{\rm limit}_{h\rightarrow 0} \frac{e^h -1}{h}\)

Question 13.51

Compute the following limits:

• (a) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,1)} e^xy\)
• (b) \(\displaystyle\mathop{\rm limit}_{x\rightarrow 0} \frac{\sin^2 x}{x}\)
• (c) \(\displaystyle\mathop{\rm limit}_{x\rightarrow 0} \frac{\sin^2 x}{x^2}\)

Question 13.52

Compute the following limits:

• (a) \(\displaystyle\mathop{\rm limit}_{x\rightarrow 3} (x^2-3x+5)\)
• (b) \(\displaystyle\mathop{\rm limit}_{x\rightarrow 0} \sin x\)
• (c) \(\displaystyle\mathop{\rm limit}_{h\rightarrow 0} \frac{(x+h)^2-x^2}{h}\)

Question 13.53

Let \[ f(x,y)= \left\{ \begin{array}{cc} \frac{xy^3}{x^2+y^6} \, & \hbox{if } (x,y)\neq (0,0) \\ 0 \, & \hbox{if } (x,y)=(0,0). \\ \end{array} \right. \]

• (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 \((0, 0)\).

Question 13.54

Let \(f(x, y, z)=\displaystyle \frac{e^{x+y}}{1+z^2}\). Compute \(\lim_{h \rightarrow 0}\frac{f(1, 2+h, 3)-f(1, 2, 3)}{h}\).

Question 13.55

Compute the following limits if they exist:

• (a) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)}\frac{(x+y)^2-(x-y)^2}{xy}\)
• (b) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{\sin xy}{y}\)
• (c) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{x^3-y^3}{x^2+y^2}\)

Question 13.56

Compute the following limits if they exist:

• (a) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{e^{xy}-1}{y}\)
• (b) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{\cos\, (xy)-1}{x^2y^2}\)
• (c) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{xy}{x^2+y^2+2}\)

Question 13.57

Compute the following limits if they exist:

• (a) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{e^{xy}}{x+1}\)
• (b) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{\cos x-1-(x^2/2)}{x^4+y^4}\)
• (c) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{(x-y)^2}{x^2+y^2}\)

Question 13.58

Compute the following limits if they exist:

• (a) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{\sin xy}{xy}\)
• (b) \(\displaystyle\mathop{\rm limit}_{(x,y,z)\,\rightarrow\, (0,0,0)} \frac{\sin\,(xyz)}{xyz}\)
• (c) \(\displaystyle\mathop{\rm limit}_{(x,y,z)\,\rightarrow\, (0,0,0)} f(x,y,z), \hbox{ where }f(x,y,z)= (x^2+3y^2)/(x+1)\)

Question 13.59

Compute the following limits if they exist:

• (a) \(\displaystyle\mathop{\rm limit}_{x\,\rightarrow\, 0} \frac{\sin 2x -2x}{x^3}\)
• (b) \(\displaystyle\mathop{\rm limit}_{(x,y)\,\rightarrow\, (0,0)} \frac{\sin 2x-2x+y}{x^3+y}\)
• (c) \(\displaystyle\mathop{\rm limit}_{(x,y,z)\,\rightarrow\, (0,0,0)} \frac{2x^2y\cos z}{x^2+y^2}\)

Question 13.60

Compute \({\rm limit}_{{\bf x}\to {\bf x}_0} f({\bf x})\), if it exists, for the following cases:

• (a) \(f\colon\, {\mathbb R}\rightarrow {\mathbb R},x\mapsto |x|,x_0=1\)
• (b) \(f\colon\, {\mathbb R}^n\rightarrow {\mathbb R}, {\bf x} \mapsto \| {\bf x} \|,\hbox{ arbitrary } {\bf x}_0\)
• (c) \(f\colon\, {\mathbb R}\rightarrow {\mathbb R}^2,x\mapsto (x^2,e^x),x_0=1\)
• (d) \(f\colon\, {\mathbb R}^2 \backslash \{(0,0)\}\rightarrow {\mathbb R}^2,(x,y)\mapsto (\sin\,(x-y), e^{x(y+1)}-x-1)/ \| (x,y) \| ,{\bf x}_0=(0,0)\)

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Question 13.61

Let \(f(x, y, z)= \frac{1}{x^2+y^2+z^2-1}\). Describe geometrically the set in \(\mathbb{R}^3\) where \(f\) fails to be continuous.

Question 13.62

Where is the function \(f(x, y)=\frac{1}{x^2+y^2}\) continuous?

Question 13.63

Let \(A= \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right]\).

• (a) Considering \(A\colon \mathbb{R}^2 \to \mathbb{R}^2\) as a linear map, explicitly write the component functions of \(A\).
• (b) Show that \(A\) is continuous on all of \(\mathbb{R}^2\).

Question 13.64

Find \(\displaystyle \lim_{(x, y) \rightarrow (0, 0)} (3x^2+3y^2)\log (x^2+y^2)\). (HINT: Use polar coordinates.)

Question 13.65

\(A=\{ (x,y) \mid -1<x<1,-1<y<1 \}\)

Question 13.66

\(B=\{(x,y)\mid y > 0\}\)

Question 13.67

\(C=\{(x,y)\mid 2 < x^2+y^2<4\}\)

Question 13.68

\(D=\{(x,y)\mid x\not=0 \hbox{ and } y\not=0\}\)

Question 13.69

Let \(A\subset {\mathbb R}^2\) be the open unit disk \(D_1(0,0)\) with the point \({\bf x}_0=(1,0)\) added, and let \(f\colon\, A\rightarrow {\mathbb R},{\bf x}\mapsto f({\bf x})\) be the constant function \(f({\bf x})=1\). Show that \({\rm limit}_{{\bf x}\to {\bf x}_0} f({\bf x})=1\).

Question 13.70

If \(f\colon\, {\mathbb R}^n\rightarrow {\mathbb R}\) and \(g\colon\, {\mathbb R}^n\rightarrow {\mathbb R}\) are continuous, show that the functions \[ f^2 g\colon\, {\mathbb R}^n\rightarrow {\mathbb R},{\bf x}\mapsto[f({\bf x})]^2g({\bf x}) \] and \[ f^2+g\colon\, {\mathbb R}^n\rightarrow {\mathbb R},x\mapsto[f({\bf x})]^2+g({\bf x}) \] are continuous.

Question 13.71

• (a) Show that \(f\colon\, {\mathbb R}\rightarrow {\mathbb R},x\mapsto (1-x)^8+\cos\,(1+x^3)\) is continuous.
• (b) Show that the map \(f\colon \,{\mathbb R} \rightarrow {\mathbb R},x\mapsto x^2e^x/(2-\sin x)\) is continuous.

Question 13.72

• (a) Can \([\sin\, (x+y)]/(x+y)\) be made continuous by suitably defining it at (0, 0)?
• (b) Can \(xy/(x^2+y^2)\) be made continuous by suitably defining it at (0, 0)?
• (c) Prove that \(f\colon\, {\mathbb R}^2\rightarrow {\mathbb R},(x,y)\mapsto ye^x+\sin x+(xy)^4\) is continuous.

Question 13.73

Using either \(\varepsilon\)’s and \(\delta\)’s or spherical coordinates, show that \[ \displaystyle\mathop{\rm limit}_{(x,y,z)\,\rightarrow\, (0,0,0)}\,\frac{xyz}{x^2+y^2+z^2}=0. \]

Question 13.74

Use the \(\varepsilon\)-\(\delta\) formulation of limits to prove that \(x^2\rightarrow 4\) as \(x\rightarrow 2\). Give another proof using Theorem 3.

Question 13.75

• (a) Prove that for \({\bf x}\in {\mathbb R}^n\) and \(s<t, D_s({\bf x})\subset D_t({\bf x})\).
• (b) Prove that if \(U\) and \(V\) are neighborhoods of \({\bf x}\in {\mathbb R}^n\), 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 \(b\).

Question 13.76

Suppose \({\bf x}\) and \({\bf y}\) are in \({\mathbb R}^n\) and \({\bf x}\not={\bf y}\). Show that there is a continuous function \(f\colon\, {\mathbb R}^n\rightarrow {\mathbb R}\) with \(f({\bf x})=1, f({\bf y})= 0\), and \(0\leq f({\bf z})\leq 1\) for every \({\bf z}\) in \({\mathbb R}^n\).

Question 13.77

Let \(f\colon\, A\subset {\mathbb R}^n\rightarrow {\mathbb R}\) be given and let \({\bf x}_0\) be a boundary point of \(A\). We say that \({\rm limit}_{{\bf x}\rightarrow {\bf x}_0} f({\bf x})=\infty\) if for every \(N>0\) there is a \(\delta >0\) such that \(0< \| {\bf x} - {\bf x}_0 \| <\delta\) and \({\bf x}\in A\) implies \(f({\bf x})>N\).

• (a) Prove that \({\rm limit}_{x\rightarrow 1}\,(x-1)^{-2}=\infty\).
• (b) Prove that \({\rm limit}_{x\rightarrow 0} 1/|x|=\infty\). Is it true that \({\rm limit}_{x\rightarrow 0} 1/x=\infty\)?
• (c) Prove that \({\rm limit}_{(x,y) \,\to\, (0, 0)} 1/ (x^2 + y^2) = \infty\).

Question 13.78

Let \(b\in{\mathbb R}\) and \(f\colon\, {\mathbb R}\backslash [b]\rightarrow {\mathbb R}\) be a function. We write \({\rm limit}_{x\rightarrow b-}f(x)=L\) 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 \(0<|x-b|<\delta\) implies \(|f(x)-L| <\varepsilon\).

• (a) Formulate a definition of right-hand limit, or \({\rm limit}_{x\rightarrow b+} f(x)\).
• (b) Find \({\rm limit}_{x\rightarrow 0-}1/(1+e^{1/x})\) and \({\rm limit}_{x\rightarrow 0+}1/(1+e^{1/x})\).
• (c) Sketch the graph of \(1/(1+e^{1/x})\).

Question 13.79

Show that \(f\) is continuous at \({\bf x}_0\) if and only if \[ \mathop {\hbox{limit }}_{{\bf x}\rightarrow {\bf x}_0} \| f({\bf x})-f({\bf x}_0) \| =0. \]

Question 13.80

Let \(f\colon\, A\subset {\mathbb R}^n\rightarrow {\mathbb R}^m\) satisfy \(\| f({\bf x})- f({\bf y}) \| \leq K \| {\bf x}-{\bf y} \| ^\alpha\) for all \({\bf x}\) and \({\bf y}\) in \(A\) for positive constants \(K\) and \(\alpha\). Show that \(f\) is continuous. (Such functions are called Hölder-continuous or, if \(\alpha=1\), Lipschitz-continuous.)

105

Question 13.81

Show that \(f\colon \,{\mathbb R}^n\rightarrow {\mathbb R}^m\) is continuous at all points if and only if the inverse image of every open set is open.

Question 13.82

• (a) Find a specific number \(\delta >0\) such that if \(|a|<\delta\), then \(|a^3+3a^2+a|<1/100\).
• (b) Find a specific number \(\delta>0\) such that if \(x^2+y^2<\delta^2\), then \[ |x^2+y^2+3xy+180xy^5|<{1/\hbox{10,000}}. \]