# Tema 8
08-3  La regla de l'Hôpital

## Indeterminaciones 0/0

kill(all)$

Ejemplo 1

f1(x):=sin(2*x);        f2(x):=3*x-x^2;

'limit(f1(x), x, 0) = limit(f1(x), x, 0);     
'limit(f2(x), x, 0) = limit(f2(x), x, 0);

define ( Df1(x) , diff(f1(x),x) );
define ( Df2(x) , diff(f2(x),x) );

'limit(Df1(x)/Df2(x), x, 0) = limit(Df1(x)/Df2(x), x, 0);

'limit(f1(x)/f2(x), x, 0)=limit(f1(x)/f2(x), x, 0);

Ejemplo 2

kill(all)$

f1(x):=1-cos(x);        f2(x):=exp(x)+exp(-x)-2;

'limit(f1(x), x, 0) = limit(f1(x), x, 0);     
'limit(f2(x), x, 0) = limit(f2(x), x, 0);

define ( Df1(x) , diff(f1(x),x) );
define ( Df2(x) , diff(f2(x),x) );

'limit(Df1(x), x, 0) = limit(Df1(x), x, 0);     
'limit(Df1(x), x, 0) = limit(Df2(x), x, 0);

define ( D2f1(x) , diff(f1(x),x,2) );
define ( D2f2(x) , diff(f2(x),x,2) );

'limit(D2f1(x), x, 0) = limit(D2f1(x), x, 0);     
'limit(D2f2(x), x, 0) = limit(D2f2(x), x, 0);

'limit(D2f1(x)/D2f2(x), x, 0) = limit(D2f1(x)/D2f2(x), x, 0);

'limit(f1(x)/f2(x), x, 0) = limit(f1(x)/f2(x), x, 0);

Ejemplo 3

kill(all)$

f1(x):=x^2*sin(1/x);  f2(x):=exp(x)-1;

'limit(f1(x)/f2(x), x, 0) = limit(f1(x)/f2(x), x, 0);

wxplot2d([f1(x)/f2(x)], [x,-0.4,0.4], [y, -0.4, 0.4], [ylabel, ""] , [legend, ""] , [style, [lines, 2]] ,
[gnuplot_preamble, "set grid"], [nticks, 300])$

## Indeterminaciones infinito/infinito

kill(all)$

f1(x):=log(x^2 + 1);     
f2(x):=x^3 + 2*x^2 + 4*x + 3;

'limit(f1(x), x, inf)=limit(f1(x), x, inf);
'limit(f2(x), x, inf)=limit(f2(x), x, inf);

define ( Df1(x) , diff(f1(x),x) );
define ( Df2(x) , diff(f2(x),x) );

'limit(Df1(x), x, inf) = limit(Df1(x), x, inf);
'limit(Df2(x), x, inf) = limit(Df2(x), x, inf);

'limit(Df1(x)/Df2(x), x, inf) = limit(Df1(x)/Df2(x), x, inf);

'limit(f1(x)/f2(x), x, inf) = limit(f1(x)/f2(x), x, inf);