Limites et asymptotes – exercices de maths Terminale

Exercice 1Usuelles

1. \(\lim_{+\infty} 2x^4 - x\)
2. \(\lim_{-\infty} \dfrac{3}{x^3}\)
3. \(\lim_{0^-} \dfrac{1}{x}\)
4. \(\lim_{-\infty} e^{x}\)
5. \(\lim_{+\infty} \sqrt{2x}\)

Exercice 2Polynômes et rationnelles

1. \(f(x) = -3x^3 + 2x - 5\)
2. \(g(x) = \dfrac{2x^2 - 1}{x^2 + 5}\)
3. \(h(x) = \dfrac{x + 4}{x^2 - 1}\)
4. \(k(x) = \dfrac{x^3}{2x - 1}\)

En \(\pm\infty\).

Exercice 3En un point

1. \(\lim_{3^\pm} \dfrac{2}{x - 3}\)
2. \(\lim_{2} \dfrac{x + 1}{(x - 2)^2}\)
3. \(\lim_{4} \dfrac{x^2 - 16}{x - 4}\)
4. \(\lim_{0} \dfrac{\sqrt{x + 4} - 2}{x}\)

Exercice 4Asymptotes

\(f(x) = \dfrac{3x - 2}{x + 1}\).

1. Limites en \(\pm\infty\), \(-1^\pm\) ?
2. Asymptotes ?
3. Position par rapport à l'horizontale ?

Exercice 5Exponentielle

1. \(\lim_{+\infty} \dfrac{e^x}{x^2}\)
2. \(\lim_{-\infty} x^2e^{x}\)
3. \(\lim_{+\infty} (x + 1)e^{-x}\)
4. \(\lim_{+\infty} \dfrac{2e^x + 1}{e^x - 3}\)
5. \(\lim_{0} \dfrac{e^{2x} - 1}{x}\)

Exercice 6Composées

1. \(\lim_{+\infty} \sqrt{x^2 + x}\)
2. \(\lim_{-\infty} e^{-x^2}\)
3. \(\lim_{0^-} e^{1/x}\)
4. \(\lim_{+\infty} \sqrt{\dfrac{2x + 1}{x}}\)

Exercice 7Comparaison

1. \(x - \cos x \to +\infty\) ?
2. \(\dfrac{\sin x}{x^2} \to 0\) ?
3. \(e^x \geq 1 + x\) ⟹ limite de \(e^x\) ?

Exercice 8Oblique

\(f(x) = 2x - 3 + \dfrac{5}{x + 2}\).

1. Limites ?
2. \(y = 2x - 3\) asymptote ?
3. Position ?

Exercice 9Étude

\(f(x) = \dfrac{x^2 + 1}{x - 2}\).

1. Limites ?
2. Variations ?
3. \(f(x) = x + 2 + \dfrac{5}{x - 2}\), asymptote oblique ?

Exercice 10Problème

\(f(x) = (3 - x)e^{-x}\).

1. Limites ?
2. Variations ?
3. Solutions de \(f(x) = k\) selon \(k\) ?
4. Courbe.