Fonctions trigonométriques/Exercices/Calcul de limites

Exercice 3-1

Calculer $\lim _{x\to 0}{\frac {\sin 5x}{7x}}$ et $\lim _{x\to 0}{\frac {\sin 5x}{\sin 7x}}$.

Solution

Ces deux limites valent ${\frac {5}{7}}$, puisque $\lim _{X\to 0}{\frac {\sin X}{X}}=1$.

 

Exercice 3-2

Calculer $\lim _{x\to 0}{\frac {\tan 5x}{7x}}$ et $\lim _{x\to 0}{\frac {\sin 5x}{\tan 7x}}$.

Solution

Ces deux limites valent ${\frac {5}{7}}$, puisque $\lim _{X\to 0}{\frac {\tan X}{X}}=\lim _{X\to 0}{\frac {\sin X}{X}}=1$.

 

Exercice 3-3

Calculer :

1°  $\lim _{x\to 0}{\frac {\sin ^{2}x}{\tan ^{2}x}}$ ;

2°  $\lim _{x\to 0}{\frac {\sin ^{2}x}{1-\cos x}}$.

Solution

1°  $\lim _{x\to 0}{\frac {\sin ^{2}x}{\tan ^{2}x}}=1$, puisque $\lim _{X\to 0}{\frac {\tan X}{X}}=\lim _{X\to 0}{\frac {\sin X}{X}}=1$.

2°  $\lim _{x\to 0}{\frac {\sin ^{2}x}{1-\cos x}}=2$, puisque $\lim _{X\to 0}{\frac {\sin X}{X}}=1$ et $\lim _{X\to 0}{\frac {1-\cos X}{X^{2}}}={\frac {1}{2}}$.

 

Exercice 3-4

Calculer $\lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}$, puis $\lim _{x\to 0}{\frac {\sin x-\cos x+1}{x}}$.

Solution

$\lim _{x\to 0}{\frac {1-\cos x}{x}}=0$, puisque $\lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}$. Par conséquent, $\lim _{x\to 0}{\frac {\sin x-\cos x+1}{x}}=\lim _{x\to 0}{\frac {\sin x}{x}}=1$.

 

Exercice 3-5

Calculer :

1°  $\lim _{x\to 0^{+}}{\frac {\sin 2x}{\sqrt {1-\cos x}}}$ ;

2°  $\lim _{x\to 0}{\frac {\tan x-\sin x}{x^{3}}}$.

Solution

1°  $\lim _{x\to 0}{\frac {\sqrt {1-\cos x}}{|x|}}={\sqrt {\lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}}}={\frac {1}{\sqrt {2}}}$ donc $\lim _{x\to 0^{+}}{\frac {\sin 2x}{\sqrt {1-\cos x}}}=2{\sqrt {2}}$.

2°  $\lim _{x\to 0}{\frac {\tan x-\sin x}{x^{3}}}=\lim _{x\to 0}{\frac {\tan x}{x}}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}$.

 

Exercice 3-6

Calculer :

1°  $\lim _{x\to {\frac {\pi }{2}}}\left(\sin x-1\right)\tan ^{2}x$ ;

2°  $\lim _{x\to \pi }\left(1+\cos x\right)\tan {\frac {x}{2}}$.

Solution

1°  $\lim _{h\to 0}\left(\sin \left({\frac {\pi }{2}}+h\right)-1\right)\tan ^{2}\left({\frac {\pi }{2}}+h\right)=\lim _{h\to 0}{\frac {\cos h-1}{\tan ^{2}h}}=-{\frac {1}{2}}$.

2°  $\lim _{h\to 0}\left(1+\cos \left(\pi +h\right)\right)\tan {\frac {\pi +h}{2}}=\lim _{h\to 0}{\frac {1-\cos h}{-\tan {\frac {h}{2}}}}=0$.

 

Exercice 3-7

Calculer :

1°  $\lim _{x\to a}{\frac {\cos x-\cos a}{\sin x-\sin a}}$ ;

2°  $\lim _{x\to a}{\frac {\tan x-\tan a}{\cos x-\cos a}}$ ;

3°  $\lim _{x\to {\frac {\pi }{3}}}{\frac {\sin 3x}{1-2\cos x}}$.

Solution

1°  ${\frac {\cos 'a}{\sin 'a}}=-\tan a$.

2°  ${\frac {\tan 'a}{\cos 'a}}=-{\frac {1}{\cos ^{2}a\sin a}}$.

3°  ${\frac {3\cos 3{\frac {\pi }{3}}}{2\sin {\frac {\pi }{3}}}}={\frac {-3}{\sqrt {3}}}=-{\sqrt {3}}$.