# 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 !)