Exercice 1
Calculer :
- \(\lim_{x\to+\infty}(2x^3 - 5x^2 + 1)\)
- \(\lim_{x\to-\infty}\frac{3x^2 - x}{1 - 2x^2}\)
- \(\lim_{x\to+\infty}\frac{x + 1}{x^2 + 3}\)
Correction
- \(+\infty\).
- \(\frac{3x^2}{-2x^2} = -\frac32\).
- \(\frac{x}{x^2} = \frac1x\to0\).
Exercice 2
Calculer :
- \(\lim_{x\to1}\frac{x^2 + x - 2}{x - 1}\)
- \(\lim_{x\to3}\frac{\sqrt{x + 1} - 2}{x - 3}\)
Correction
- \(\frac{(x - 1)(x + 2)}{x - 1} = x + 2\to3\).
- Conjuguée : \(\frac{x + 1 - 4}{(x - 3)(\sqrt{x+1} + 2)} = \frac{1}{\sqrt{x+1} + 2}\to\frac14\).
Exercice 3
Calculer \(\lim_{x\to+\infty}(\sqrt{x^2 + x} - x)\).
Correction
\(\frac{x^2 + x - x^2}{\sqrt{x^2 + x} + x} = \frac{x}{x(\sqrt{1 + \frac1x} + 1)}\to\frac12\).
Exercice 4
Calculer : \(\lim_{x\to0}\frac{\sin 3x}{x}\), \(\lim_{x\to0}\frac{1 - \cos x}{x\sin x}\), \(\lim_{x\to0}\frac{\tan 2x}{\sin 5x}\).
Correction
\(3\cdot\frac{\sin 3x}{3x}\to3\). \(\frac{1 - \cos x}{x^2}\cdot\frac{x}{\sin x}\to\frac12\). \(\frac{\tan 2x}{2x}\cdot\frac{5x}{\sin 5x}\cdot\frac25\to\frac25\).
Exercice 5
Calculer \(\lim_{x\to+\infty}\frac{\sin x}{x}\).
Correction
\(\left|\frac{\sin x}{x}\right|\le\frac1x\) pour \(x > 0\) et \(\frac1x\to0\) : la limite est \(0\).
Exercice 6
Calculer \(\lim_{x\to2^+}\frac{x + 1}{x - 2}\) et \(\lim_{x\to2^-}\frac{x + 1}{x - 2}\).
Correction
Numérateur \(\to3\). Pour \(x\to2^+\), \(x - 2\to0^+\) : \(+\infty\). Pour \(x\to2^-\) : \(-\infty\).