تمارين - 1BACSEF

الأولى باكالوريا العلوم التجريبية – خيار فرنسي

درس : Limite d'une fonction numérique

Exercice 12

Calculer les limites suivantes :

  • lim⁡x→+∞x+2xx−1\lim\limits_{x\to +\infty} \dfrac{x+2\sqrt{x}}{x-1}

  • lim⁡x→2x2−xx+x\lim\limits_{x\to 2} \dfrac{x^2-x}{x+\sqrt{x}}

lim⁡x→+∞x+2xx−1=lim⁡x→+∞x(1+2xx)x(1−1x)=lim⁡x→+∞1+2xx21−1x=lim⁡x→+∞1+21x1−1x=1\begin{align*} \lim\limits_{x\to +\infty} \dfrac{x+2\sqrt{x}}{x-1} &=\lim\limits_{x\to +\infty} \dfrac{x\left(1+2\dfrac{\sqrt x}x\right)}{x\left(1-\dfrac1x\right)} \\ &=\lim\limits_{x\to +\infty} \dfrac{1+2\sqrt\dfrac{x}{x^2}}{1-\dfrac1x}\\ &=\lim\limits_{x\to +\infty} \dfrac{1+2\sqrt\dfrac{1}x}{1-\dfrac1x}\\ &=1 \end{align*}

car lim⁡x→+∞1x=0\lim\limits_{x\to +\infty} \dfrac1x=0

lim⁡x→2x2−xx+x=22−22+2=22+2=2−2\begin{align*} \lim\limits_{x\to 2} \dfrac{x^2-x}{x+\sqrt{x}} &= \dfrac{2^2-2}{2+\sqrt{2}}=\dfrac{2}{2+\sqrt{2}}\\ &=2-\sqrt2 \end{align*}