exo10-11-12.mw

>

> Exercice 10

Question a)

> f:=x->sin(x)-x:f(x)=series(f(x),x=0,3);

sin(x)-x = (series(+O(x^3),x,3))

> Limit(f(x)/x^2,x=0)=limit(f(x)/x^2,x=0);

Limit((sin(x)-x)/x^2, x = 0) = 0

> Question b)

> f:=x->sin(x)-x:f(x)=series(f(x),x=0,4);

sin(x)-x = (series(-1/6*x^3+O(x^5),x,5))

> g:=x->x*ln(1-x^2):g(x)=series(g(x),x=0,4);

x*ln(1-x^2) = (series(-x^3+O(x^5),x,5))

> Limit(f(x)/g(x),x=0)=limit(f(x)/g(x),x=0);

Limit((sin(x)-x)/(x*ln(1-x^2)), x = 0) = 1/6

> Question c)

> f:=x->ln(sin(x)/x):f(x)=series(f(x),x=0,5);

ln(sin(x)/x) = (series(-1/6*x^2-1/180*x^4+O(x^5),x,5))

> g:=x->1/f(x):g(x)=series(g(x),x=0,5);

1/ln(sin(x)/x) = (series(-6*x^(-2)+1/5+O(x^1),x,1))

> Limit(1/x-g(x),x=0)=limit(1/x-g(x),x=0);

Limit(1/x-1/ln(sin(x)/x), x = 0) = infinity

> Question d)

> f:=x->sqrt(sin(x)/x)-1:f(x)=series(f(x),x=0,4);

(sin(x)/x)^(1/2)-1 = (series(-1/12*x^2+O(x^4),x,4))

> g:=x->sin(x)/x-1:g(sqrt(x))=subs(x=sqrt(x),series(g(x),x=0,3));

sin(x^(1/2))/x^(1/2)-1 = -1/6*x+O(x^2)

> Limit(f(x)/g(sqrt(x)),x=0)=limit(f(x)/g(x),x=0);

Limit(((sin(x)/x)^(1/2)-1)/(sin(x^(1/2))/x^(1/2)-1), x = 0) = 1/2

>

> Exercice 11

>

> f:=x->arctan(x)-sin(x):f(x)=series(f(x),x=0,4);

arctan(x)-sin(x) = (series(-1/6*x^3+O(x^4),x,4))

> g:=x->tan(x)-arcsin(x):g(x)=series(g(x),x=0,4);

tan(x)-arcsin(x) = (series(1/6*x^3+O(x^4),x,4))

> Limit(f(x)/g(x),x=0)=limit(f(x)/g(x),x=0);

Limit((arctan(x)-sin(x))/(tan(x)-arcsin(x)), x = 0) = -1

>

> Exercice 12

> Question a)

> f:=x->exp(x)-(cos(x)+x):f(x)=series(f(x),x=0,3);

exp(x)-cos(x)-x = (series(x^2+O(x^3),x,3))

> Limit(f(x)/x^2,x=0)=limit(f(x)/x^2,x=0);

Limit((exp(x)-cos(x)-x)/x^2, x = 0) = 1

> Question b)

> f:=x->x^3*arctan(x)-x^4:f(x)=series(f(x),x=0,7);

x^3*arctan(x)-x^4 = (series(-1/3*x^6+O(x^8),x,8))

> g:=x->cos(x^2)-1:g(x)=series(g(x),x=0,5);

cos(x^2)-1 = (series(-1/2*x^4+O(x^7),x,7))

> Limit(f(x)/g(x),x=0)=limit(f(x)/g(x),x=0);

Limit((x^3*arctan(x)-x^4)/(cos(x^2)-1), x = 0) = 0

>