Séance 1-1-2 : Limites et continuité - Partie 1 (Exercices)

Sommaire

II- Exercices I

2-1/ Exercice 1-1

2-2/ Exercice 1-2

2-3/ Exercice 1-3

2-4/ Exercice 1-4

II- Exercices I

2-1/ Exercice 1-1

Calculer les limites suivantes :

$$\begin{gathered} \overline{)1} \underset{x\to 4}{\lim }\frac{\left|x^2-2x\right|-8}{x^2-5x+4} \\ \overline{)2} \underset{x\to +\infty }{\lim }\frac{\sqrt{x^2+x+1}-\sqrt{x^2-4}}{x+\sqrt{x+2}} \\ \overline{)3} \underset{x\to -\infty }{\lim }\sqrt{x^4-x^3}-x^2 \\ \overline{)4} \underset{x\to +\infty }{\lim }x\sqrt{4x^2+3x-7}-2x^2 \end{gathered}$$

II- Exercices I

2-2/ Exercice 1-2

Calculer les limites suivantes :

$$\begin{gathered} \overline{)1} \underset{x\to -\infty }{\lim }\frac{\sqrt{10x^2+9}-7}{\sqrt{2-x}+\sqrt{x^2+5}-5} \\ \overline{)2} \underset{x\to -\infty }{\lim }\frac{\sqrt{x^4+1}-\sqrt{x^4-1}}{\sqrt{x^2+1}-\sqrt{x^2-1}} \\ \overline{)3} \underset{x\to +\infty }{\lim }\sqrt{x^3+\alpha x^2+x+1} \left(\alpha \in \mathrm{ℝ}\right) \\ \overline{)4} \underset{x\to {\alpha }^{+}}{\lim }\frac{\sqrt{x^2-\alpha x}+\sqrt{x^2-{\alpha }^2}}{\sqrt{x-\alpha }} \left(\alpha \in \mathrm{ℝ}\right) \end{gathered}$$

II- Exercices I

2-3/ Exercice 1-3

Calculer les limites suivantes :

$$\begin{gathered} \overline{)1} \underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{x+x\cos 4x}{\sin x-\cos x } \\ \overline{)2} \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{2\tan 2x-\sqrt{3}\cos x}{\mathrm{\pi }-2x} \\ \overline{)3} \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left(x\tan x-\frac{\mathrm{\pi }}{2\cos x}\right) \\ \overline{)4} \underset{x\to \frac{\mathrm{\pi }}{3}}{\lim }\frac{\sqrt{3}\sin x-\cos x-1}{2\cos x+\cos 3x} \end{gathered}$$

II- Exercices I

2-4/ Exercice 1-4

Calculer les limites suivantes :

$$\begin{gathered} \overline{)1} \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left(\frac{\mathrm{\pi }}{2}-x\right)\tan x \\ \overline{)2} \underset{x\to 1}{\lim }\left(2-3x+x^2\right)\tan \left(\frac{\mathrm{\pi }}{2}x\right) \\ \overline{)3} \underset{x\to +\infty }{\lim }\left(x^2+x-1\right){\sin }^2\left(\frac{1}{x}\right) \\ \overline{)4} \underset{x\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{{\cos }^{3}x-{\sin }^{3}x}{\sin \left(\frac{\mathrm{\pi }}{4}-x\right)} \\ \overline{)5} \underset{x\to 0}{\lim }\frac{\cos ax-\cos bx}{x^2} \left(\left(a,b\right)\in {\mathrm{ℝ}}^2\right) \end{gathered}$$