printf(" \n\n"
 "With the computation we arrive at the "
 "following conjecture : b = %+.1f \n\n", 
 (*P_f)(i*1000000000)); 
 stop();
 return 0;
/* ------------------------------------ */
Exemple de sortie écran :
 The curve y = ax**2 + bx + c (a!=0) 
 is a branch of the curve of f at +oo if 
 lim f(x) 
 x->+oo ---- = a 
 x**2 
 lim [f(x)-ax**2] 
 and x->+oo --------- = b 
 x 
 lim [f(x)-ax**2-bx] = c 
 and x->+oo 
 Press return to continue. 
 f : x-> (-2*x**3 - 3*x**2 + 2*x + 5)/(2*x+3)
 Compute : lim [f(x)-ax**2] 
 x->+oo --------- = b (with a=1) 
 x 
 L2(1e+02) = 1e-02 || L2(1e+09) = 0e+00
 L2(2e+02) = 5e-03 || L2(2e+09) = 0e+00
 L2(3e+02) = 3e-03 || L2(3e+09) = -3e-07
 L2(4e+02) = 3e-03 || L2(4e+09) = 0e+00
 L2(5e+02) = 2e-03 || L2(5e+09) = 0e+00
 L2(6e+02) = 2e-03 || L2(6e+09) = 0e+00
 L2(7e+02) = 1e-03 || L2(7e+09) = 0e+00
 L2(8e+02) = 1e-03 || L2(8e+09) = 0e+00
 L2(9e+02) = 1e-03 || L2(9e+09) = 2e-06
With the computation we arrive at the following conjecture : b = +0.0 
 Press return to continue.

Mathc initiation/c76d
Sommaire 
Installer et compiler ces fichiers dans votre répertoire de travail.
/* Save as : c01d.c */
int main(void)
double i;
double (*P_f)(double x);
 P_f = fc;
 clrscrn();
 printf(" The curve y = ax**2 + bx + c (a!=0) \n\n\n" 
 " is a branch of the curve of f at +oo if \n\n\n\n" 
 " lim f(x) \n" 
 " x->+oo ---- = a \n" 
 " x**2 \n\n\n" 
 " lim [f(x)-ax**2] \n" 
 " and x->+oo --------- = b \n" 
 " x \n\n\n" 
 " lim [f(x)-ax**2-bx] = c \n" 
 " and x->+oo \n\n\n"); 
 stop();
 clrscrn();
 printf(" f : x-> %s\n\n\n", feq);
 printf(" Compute : lim [f(x)-ax**2-bx] = c (with a=1 b=0) \n");
 printf(" x->+oo \n\n\n");
 for(i=1; i<10; i+=1)
 printf(" L3(%.0e) = %-10.0e || L3(%.0e) = %.2e\n",
 i*100, (*P_f)(i*100),
 i*10000, (*P_f)(i*10000) );

printf(" \n\n"
 "With the computation we arrive at the "
 "following conjecture : c = %+.1f \n\n", 
 (*P_f)(i*10000)); 
 stop();
 return 0;
/* ------------------------------------ */
Exemple de sortie écran :
 The curve y = ax**2 + bx + c (a!=0) 
 is a branch of the curve of f at +oo if 
 lim f(x) 
 x->+oo ---- = a 
 x**2 
 lim [f(x)-ax**2] 
 and x->+oo --------- = b 
 x 
 lim [f(x)-ax**2-bx] = c 
 and x->+oo 
 Press return to continue. 
 f : x-> (-2*x**3 - 3*x**2 + 2*x + 5)/(2*x+3)
 Compute : lim [f(x)-ax**2-bx] = c (with a=1 b=0) 
 x->+oo 
 L3(1e+02) = 1e+00 || L3(1e+04) = 1.00e+00
 L3(2e+02) = 1e+00 || L3(2e+04) = 1.00e+00
 L3(3e+02) = 1e+00 || L3(3e+04) = 1.00e+00
 L3(4e+02) = 1e+00 || L3(4e+04) = 1.00e+00
 L3(5e+02) = 1e+00 || L3(5e+04) = 1.00e+00
 L3(6e+02) = 1e+00 || L3(6e+04) = 1.00e+00
 L3(7e+02) = 1e+00 || L3(7e+04) = 1.00e+00
 L3(8e+02) = 1e+00 || L3(8e+04) = 1.00e+00
 L3(9e+02) = 1e+00 || L3(9e+04) = 1.00e+00
With the computation we arrive at the following conjecture : c = +1.0 
 Press return to continue.

Mathc initiation/c76e
Sommaire 
Installer et compiler ces fichiers dans votre répertoire de travail.
/* Save as : c01e.c */
int main(void)
double (*P_fax2)(double x);
double (*P_fbx) (double x);
double (*P_fc) (double x);
 P_fax2 = fax2;
 P_fbx = fbx;
 P_fc = fc;
 clrscrn();
 printf(" The curve y = ax**2 + bx + c (a!=0) \n\n\n" 
 " is a branch of the curve of f at +oo if \n\n\n\n" 
 " lim f(x) \n" 
 " x->+oo ---- = a \n" 
 " x**2 \n\n\n" 
 " lim [f(x)-ax**2] \n" 
 " and x->+oo --------- = b \n" 
 " x \n\n\n" 
 " lim [f(x)-ax**2-bx] = c \n" 
 " and x->+oo \n\n\n"); 
 stop();
 clrscrn(); 
 printf(" f : x-> %s\n\n\n", feq);
 printf(" lim f(x) \n");
 printf(" a = x->+oo ---- = %.8f\n", (*P_fax2)(10000)); 
 printf(" x**2 \n\n\n");
 printf(" b = lim [f(x)-a*x**2] = %.8f\n", (*P_fbx)(100000000)); 
 printf(" x->+oo \n\n\n");

printf(" c = lim [f(x)-ax**2-bx] = %.8f\n",(*P_fc)(10000)); 
 printf(" x->+oo \n\n\n");
 printf(" The curve y = %.1f x**2 %+.1f x %+.1f",
 (*P_fax2)(10000),(*P_fbx)(10000000),(*P_fc)(10000));
 printf(" is an asymptote at the curve of f at +oo \n\n\n"); 
 stop();
 return 0;
/* ------------------------------------ */
 Fichier de commande gnuplot :
set zeroaxis lt 3 lw 1
 plot[-5.:5.] [-20.:5.]\
 (-2*x**3 - 3*x**2 + 2*x + 5)/(2*x+3),\
 -x**2+1 
 reset
Exemple de sortie écran :
 The curve y = ax**2 + bx + c (a!=0) 
 is a branch of the curve of f at +oo if 
 lim f(x) 
 x->+oo ---- = a 
 x**2 
 lim [f(x)-ax**2] 
 and x->+oo --------- = b 
 x 
 lim [f(x)-ax**2-bx] = c 
 and x->+oo 
 Press return to continue. 
 f : x-> (-2*x**3 - 3*x**2 + 2*x + 5)/(2*x+3)
 lim f(x) 
 a = x->+oo ---- = -0.99999999
 x**2 
 b = lim [f(x)-a*x**2] = 0.00000002
 x->+oo 
 c = lim [f(x)-ax**2-bx] = 1.00009999
 x->+oo 
 The curve y = -1.0 x**2 +0.0 x +1.0 is an asymptote at the curve of f at +oo 
 Press return to continue.

Mathc initiation/c770
On utilise le terme de branche quand l'asymptote est une courbe.
<br>
Les fonctions pour les différents exemples :
<br>
Calculer la branche de l'équation de la forme y=ax**+bx+c :

Mathc initiation/h77a
Sommaire 
Installer ce fichier dans votre répertoire de travail.
/* Save as : fa.h */
double f(
double x)
 return( (pow(x,6)+pow(x,4)) / (pow(x,4)-x-1) +10*x );
char feq[] = "((x**6+x**4)/(x**4-x-1)) + 10*x";
double fax2(
double x)
 return((
 (pow(x,6) + pow(x,4)) 
 (pow(x,4) - x-1) 
 + 10*x 
 / pow(x,2) );
char faxeq[] = "(((x**6+x**4)/(x**4-x-1)) + 10*x) / x**2";
double fbx(
double x)
 return((
 ( 
 (pow(x,6) + pow(x,4)) 
 (pow(x,4) - x-1) 
 + 10*x 
 - pow(x,2) ) 
 / x );
char fbxeq[] = "((((x**6+x**4)/(x**4-x-1)) + 10*x)-x**2) / x";
double fc(
double x)
 return((
 ( 
 (pow(x,6) + pow(x,4)) 
 (pow(x,4) - x-1.) 
 + 10.*x 
 - pow(x,2) - 10.*x ) 
char fceq[] = "((((x**6+x**4)/(x**4-x-1)) + 10*x)-x**2-10*x)";
double fax2bxc(
double x)
 return( pow(x,2)+10*x+1 );
char fax2bxceq[] = "x**2+10*x+1";
/* ------------------------------------ */
 Déclaration des fichiers h.

Mathc initiation/c77b
Sommaire 
Installer et compiler ces fichiers dans votre répertoire de travail.
/* Save as : c01b.c */
int main(void)
double i;
double (*P_f)(double x);
 P_f = fax2;
 clrscrn();
 printf(" The curve y = ax**2 + bx + c (a!=0) \n\n\n" 
 " is a branch of the curve of f at +oo if \n\n\n\n" 
 " lim f(x) \n" 
 " x->+oo ---- = a \n" 
 " x**2 \n\n\n" 
 " lim [f(x)-ax**2] \n" 
 " and x->+oo --------- = b \n" 
 " x \n\n\n" 
 " lim [f(x)-ax**2-bx] = c \n" 
 " and x->+oo \n\n\n"); 
 stop();
 clrscrn();
 printf(" f : x-> %s\n\n\n", feq);
 printf(" lim f(x) \n");
 printf(" Compute : x->+oo ---- = a \n");
 printf(" x**2 \n\n\n");
 for(i=1; i<10; i+=1)
 printf(" L1(%.0e) = %-10.0e || L1(%.0e) = %.0e\n",
 i*100, (*P_f)(i*100),
 i*1000000000, (*P_f)(i*1000000000) );

printf(" \n\n"
 "With the computation we arrive at the "
 "following conjecture : a = %+.1f \n\n", 
 (*P_f)(i*1000000000)); 
 stop();
 return 0;
/* ------------------------------------ */
Exemple de sortie écran :
 The curve y = ax**2 + bx + c (a!=0) 
 is a branch of the curve of f at +oo if 
 lim f(x) 
 x->+oo ---- = a 
 x**2 
 lim [f(x)-ax**2] 
 and x->+oo --------- = b 
 x 
 lim [f(x)-ax**2-bx] = c 
 and x->+oo 
 Press return to continue. 
 f : x-> ((x**6+x**4)/(x**4-x-1)) + 10*x
 lim f(x) 
 Compute : x->+oo ---- = a 
 x**2 
 L1(1e+02) = 1e+00 || L1(1e+09) = 1e+00
 L1(2e+02) = 1e+00 || L1(2e+09) = 1e+00
 L1(3e+02) = 1e+00 || L1(3e+09) = 1e+00
 L1(4e+02) = 1e+00 || L1(4e+09) = 1e+00
 L1(5e+02) = 1e+00 || L1(5e+09) = 1e+00
 L1(6e+02) = 1e+00 || L1(6e+09) = 1e+00
 L1(7e+02) = 1e+00 || L1(7e+09) = 1e+00
 L1(8e+02) = 1e+00 || L1(8e+09) = 1e+00
 L1(9e+02) = 1e+00 || L1(9e+09) = 1e+00
With the computation we arrive at the following conjecture : a = +1.0 
 Press return to continue.