Calcul de limites

Exercice 1051. Calculer $\displaystyle \lim_{x \to \frac{\pi}{2}} \Frac{\ln(\sin^2(x))}{\parenthese{\Frac{\pi}{2}-x}^2}$.
Exercice 1052. Calculer $\displaystyle \lim_{x \to +\infty} \parenthese{\Frac{\ln(x+1)}{\ln(x)}}^{x\ln(x)}$.
Exercice 1053. Calculer $\displaystyle \lim_{x \to 0} \parenthese{\Frac{x}{\sin(x)}}^{\frac{\sin(x)}{x-\sin(x)}}$.
Exercice 1054. Déterminer les limites suivantes, lorsque celles-ci existent : \\
  1. $\lim\limits_{x \to 0} \Frac{\sqrt{1+x} - \sqrt{1-x}}{x}$ \\
  2. $\lim\limits_{x \to +\infty} \Frac{x - \sqrt{x}}{\ln x + x}$ \\
  3. $\lim\limits_{x \to 0^+} x^x$ \\
  4. $\lim\limits_{x \to 1^+} \ln x \; \ln(\ln x)$ \\
  5. $\lim\limits_{x \to 0} (1+x)^{1/x}$ \\
  6. $\lim\limits_{x \to 1} \Frac{1-x}{\arccos x}$
Exercice 1055. \\
  1. $\lim\limits_{x \to 0} x \sin\parenthese{\Frac{1}{x}}$ \\
  2. $\lim\limits_{x \to +\infty} \Frac{x \cos(e^x)}{x^2 + 1}$ \\
  3. $\lim\limits_{x \to +\infty} e^{x - \sin x}$ \\
  4. $\lim\limits_{x \to +\infty} \Frac{x + \arctan x}{x}$ \\
  5. $\lim\limits_{x \to 0} x \; [1/x]$ \\
  6. $\lim\limits_{x \to +\infty} x \; [1/x]$
Exercice 1056. \\
  1. $\lim\limits_{x \to 0^+} [1/x]$ \\
  2. $\lim\limits_{x \to 0} x \; [1/x]$ \\
  3. $\lim\limits_{x \to 0} x^2 \; [1/x]$