límits

lim[x-->a][ (f(x)/x) ] = 1 <==> lim[x-->a][ f(x) ] = a

lim[x-->a][ (f(x)/(-x)) ] = 1 <==> lim[x-->a][ f(x) ] = (-a)

lim[x-->a][ (f(x)·x) ] = 1 <==> lim[x-->a][ f(x) ] = (1/a)

lim[x-->a][ (f(x)·(-x)) ] = 1 <==> lim[x-->a][ f(x) ] = (1/(-a))

x·(1/x) = 1

0·(1/0) = 1 = 0·oo

x^{n}·(1/x^{n}) = 1

0^{n}·(1/0^{n}) = 1 = 0^{n}·oo^{n}

0^{n} < 0 <==> oo < oo^{n}

(-0) < (-1)·0^{n}  <==> (-1)·oo^{n} < (-1)·oo

(-0) < ...< (-1)·0^{n} < 0^{n} < ...< 0 <==> (-1)·oo^{n} < ...< (-1)·oo < oo < ...< oo^{n}

lim[ n ] = oo

lim[ (1/n) ] = 0

(n/m)·oo+1 = (n/m)·oo

1+(m/n)·0 = 1

lim[ (n^{p}+an^{p+1})/(n^{p}+bn^{p+1}) ] = ...

... lim[ (n^{p+1}/n^{p+1})·((1/n)+a)/((1/n)+b) ] = (a/b)

lim[ (n^{p}+an^{p+1})/(n^{p}+bn^{p+1}) ] = ...

... lim[ (n^{p}/n^{p})·(1+an)/(1+bn) ] = (a/b)

(n/m)·oo^{(n+1)}+oo^{n} = (n/m)·oo^{(n+1)}

0^{n}+(m/n)·0^{(n+1)} = 0^{n}

lim[ (2n)/(n+2n^{2}+3n^{3}+4n^{4}) ] = (1/2)·0^{3}

lim[ (2n+4n^{2})/(n+2n^{2}+3n^{3}+4n^{4}) ] = 0^{2}

lim[ (2n+4n^{2}+8n^{3})/(n+2n^{2}+3n^{3}+4n^{4}) ] = 2·0

lim[ (2n+4n^{2}+8n^{3}+16n^{4})/(n+2n^{2}+3n^{3}+4n^{4}) ] = 4

lim[ (1+(1/n))^{n} ] = e

d_{x}[ln(x)] = lim[h-->0][ (1/x)·ln( (1+(h/x) )^{x/h}) = (1/x)

lim[h-->0][ ( 1+(h/x) )^{x/h} ] = e

lim[ ( 1+( (kn+1)/(n^{2}+1) ) )^{n} ] = ...

... lim[ ( ( 1+( (kn+1)/(n^{2}+1) )^{(n^{2}+1)/(kn+1)} )^{( ( n(kn+1) )/(n^{2}+1) )} ] = e^{k}

x+(-x) = 0

(-x)+x = (-0)

0^{n}+(-1)·0^{n} = 0^{(n+1)}

(-1)·0^{n}+0^{n} = (-1)·0^{(n+1)}

oo^{(n+1)}+(-1)·oo^{(n+1)} = oo^{n}

(-1)·oo^{(n+1)}+oo^{(n+1)} = (-1)·oo^{n}

lim[ (n+1)^{m}+(-1)·n^{m} ] = ( m+1 )·oo^{( m+(-1) )}

lim[ (1/(n+1)^{m})+(-1)·(1/n^{m}) ] = ( (-m)+1 )·oo^{( (-m)+(-1) )}

lim[ ( (1+...(n)...+n)/n ) ] = (oo/2)

Stolz:

lim[ (n+1)/((n+1)+(-1)·n) ] = (oo/2) [ oo+(-1)·oo = 1 ]

lim[ ( (1^{2}+...(n)...+n^{2})/n^{2} ) ] = (oo/3)

Stolz:

lim[ (n+1)^{2}/((n+1)^{2}+(-1)·n^{2}) ] = (oo/3) [ oo^{2}+(-1)·oo^{2} = oo ]

lim[ ( (1^{3}+...(n)...+n^{3})/n^{3} ) ] = (oo/4)

Stolz:

lim[ (n+1)^{3}/((n+1)^{3}+(-1)·n^{3}) ] = (oo/4) [ oo^{3}+(-1)·oo^{3} = oo^{2} ]

Stolz:

Si lim[ a_{n+1} ] >] oo ==>

l+(-s) < a_{n+1}/(a_{n+1}+(-1)·a_{n}) < l+s

(a_{n+1}+(-1)·a_{1})·(l+(-s)) < a_{2}+...+a_{n+1} < (l+s)·(a_{n+1}+(-1)·a_{1})

l+(-s) < ( (a_{2}+...+a_{n+1})/a_{n+1} ) < l+s

l+(-s) < ( (a_{1}+...+a_{n})/a_{n} ) < l+s

lim[ ln(n!)/ln(n) ] = lim[ ( ln(n+1)/ln(1+(1/n)) ) ] = lim[ ln(n+1)·n ] = ln(oo^{oo})

lim[ (k+k^{2}+...+k^{n})/k^{n} ] = ...

... lim[ ( k^{(n+1)}/(k^{n+1}+(-1)·k^{n}) ) ] = ( k/(k+(-1)) )

lim[ (k+k^{(1/2)}+...+k^{(1/n)})/k^{(1/n)} ] = ...

... lim[ ( k^{( 1/(n+1) )}/(k^{( 1/(n+1) )}+(-1)·k^{(1/n)}) ) ] = ( 1/(1+(-1)) ) = oo

lim[ (1!+2!+...+n!)/n! ] = lim[ ( (n+1)!/((n+1)!+(-1)·n!) ) ] = ( (n+1)/n ) = 1

oo^{a_{1}}+...+oo^{a_{n}} = oo^{max{a_{1},...,a_{n}}}

0^{a_{1}}+...+0^{a_{n}} = 0^{min{a_{1},...,a_{n}}}