exo7 632

Calculer les limites suivantes : $$\text{a) }\ \lim_{x\rightarrow 0}\ \ \frac{x^2+2|x|}{x} \qquad \text{b) }\ \lim_{x\rightarrow 1}\ \ \frac{x^7-1}{x^6-1} \qquad \text{c) }\ \lim_{x\rightarrow 1}\ \ \frac{x^n-1}{x^m-1} \ \ n,m\in \Nn^*$$

$$\text{d) }\ \lim_{x\rightarrow 0}\ \frac{1-\cos x}{x^2} \qquad \text{e) }\ \lim_{x\rightarrow 0}\ \frac{x\sin x}{1-\cos x} \qquad \text{f) }\ \lim_{x\rightarrow 0}\ \frac{\ln (1+x^3)}{x}$$

$$\text{g) }\ \lim_{x\rightarrow 0}\ \frac{a^x-b^x}{x} \ \ a,b>0 \qquad \text{h) }\ \lim_{x\rightarrow 0}\ \frac{e^{-ax}-e^{-bx}}{x} \qquad \text{i) }\ \lim_{x\rightarrow \alpha^+}\ \frac{\sqrt{x}-\sqrt{\alpha}+\sqrt{x-\alpha}}{\sqrt{x^2-\alpha^2}}.$$