Unsual equations, limits, integrals, functions, semigroups…

In the present article we give examples of unusual equations, limits, problems, and so on…

Same example of strange equations

Solve the following equations:

1.x + 2 = 3,8
Sol
: very simple x=8 or 80 or 800,… that is 8\cdot 10^{n}, n positive integer; and don’t forget 8\cdot 10^{\infty} that is 8 with infinite zeroes, a giant number! And don’t forget 7999999… another giant number that we write 7{(9)}^{\infty}

0.x + 3 = 2,8
it’s impossible: 3.x=2.8 \Rightarrow int(3.x)=int(2.8) \Rightarrow 3=2

1.x= \sqrt{2}
Sol
: 0.x + 1 = \sqrt{2}, 0.x =\sqrt{2} -1, x=10(\sqrt{2}-1)=41…..a giant integer!

2.x+3=a
Sol
: 5.x=a if int(a) \neq 5 impossible if int a =5 0.x=a-5, x=10(a-5)10^{n}

3.x+2.y=5,6
Sol
: 3+0.x+2+0.y=5+0.6, 0.x+0,y=0.6,  one could think 10(0.x+0.y)=10 0.6, x+y=6 but look at the solution x=433, y=167: 3.433+2.167=5.6| which is then the general solution? One moment thought gives: x+y=6 or 60 or 600 that is x+y= 6\cdot 10^{n} and don’t forget x+y=6\cdot 10^{\infty}:
6\cdot 10^{\infty}=60000000…=59999999….= (e.g.)=32222…+277777.. and so we have also the solutions
x=3(2) and 2(7) two giant periodic or rational numbers!

3.x+2.y=6.3
Sol
: 0.x+0,y=1.3,here one could think as above x+y=13 but look at the solution: x=999 and y=301: 3.999+2.301=6.3 as above the complete solution is x+y=13\cdot 10^{n} together with x+y=13\cdot 10^{\infty} a sum decomposition of a giant decimal number!

1,x+0.y=\sqrt{2}=1.41…
Sol
: x+y=10(\sqrt{2}-1)=41…\cdot 10^{n} now the factor 10^{n} gives no effect because it adjoins zeroes after infinite digits with no real effect: then x+y=41…a sum decomposition of a giant irrational number!

\log _{ 3 }{ (senx } )=\log _{ 3 }{ (cosx) } +\cfrac { 1 }{ 2 }
\log _{ cosec(x) }{ cos(x)=\log _{ sen(x) }{ cosec(x) } }
sen(\pi sen(x))=0      sol    x=k\cfrac {\pi}{2}
sen(\pi cos(x))=1

Monomials:
{a}^{\sqrt{b}}{a}^{2}{({a}^{\sqrt{b}}{a}^{2}+{b}^{\sqrt{a}})}^{2}

Same example of strange limits:

Consider the following succession of functions:
{ y }_{ 1 }(x)=\left| \left|x\right| -1 \right|
{ y }_{ n+1 }(x)=\left| \left| { y }_{ n }(x) \right| -1 \right|
Calculate, if it exists, \lim_{ n\rightarrow \infty}{ { y }_{ n } }(x),
Sol: the reader is invited to draw a painting of the functions and then it’s easy to recognize that the pointwise limit is \cfrac {1}{2} for x=\cfrac {1}{2}+k, k integer and dosn’t exists elsewhere. In this second case {y}_{n}(x) is alternately x-int(x) and 1-(x-int(x)),
Calculate, if it exists, \lim _{ x\rightarrow { 0 }^{ + } }{ \log _{ x }{ { (e }^{ x }-1) } } ,
Sol: it,s in the indeterminate form \log _{ { 0 }^{ + } }{ { 0 }^{ + } } . To solve it we use the formula: \log _{ a }{ b } =\frac { \log _{ c }{ b } }{ \log _{ c }{ a } } obtaining \lim _{ x\rightarrow { o }^{ + } }{ \frac { \ln { { (e }^{ x }-1) } }{ \ln { x } } } …With this technique we can solve every limit of the form \lim _{ x\rightarrow \alpha }{ \log _{ f(x) }{ g(x) } }

Same example of strange integrals:

\int { { x }^{ x } } (lnx+1)dx:
sol: { x }^{ x }, simply derivate it
\int { { e }^{ x{ e }^{ x }+x } } (x+1)dx
sol: set { e }^{ x}= t and obtain the previous one
\int { { x }^{ { x }^{ 2 }+1 }ln(ex)dx }
sol: { x }^{ { x }^{ 2 } }
\int {sin(cos(x)) }dx

Same example of strange functions:

f(x)=\sqrt [ \sqrt { x } ]{ x }
f(x)={\pi}^{x}+{x}^{\pi}

Derivations:
\frac{{\pi}^{x}+3\pi+{x}^{\pi}+{(\pi x)}^{{x}^{2}-\pi}}{5\pi}   (exercise in classwork given by prof.ssa Orso Cristina class Va L.S:S:A.    I.T.I.S. Q.Sella Biella Italy 13/12/2017)

Integral functions:
{ F }_{ 1 }(x)=\int _{ 0 }^{ x }{ \frac { 1 }{ \sqrt [ 3 ]{ { sen }^{ 2 }(t) } } dt },   { F }_{ n }(x)=\frac { 1 }{ n } \int _{ 0 }^{ x }{ \frac { 1 }{ \sqrt [ 3 ]{ { sen }^{ 2 }(nt) } } dt },      {F}_{\infty}(x)=\lim _{ n\rightarrow \infty }{ { F }_{ n }(x) }   (if it exists !)

References:
B:R:Gelbaum, J.H.Olmsted Counterexamples in Analysis 1964 Holden-Day, Inc.