Règle de L'Hospital - Savoirs et savoir-faire

La règle de L'Hospital permet de lever des indéterminations du type $\dfrac{0}{0}$start fraction, 0, divided by, 0, end fraction ou $\dfrac{\infty}{\infty}$start fraction, infinity, divided by, infinity, end fraction.

Autrement dit, elle permet de déterminer $\displaystyle\lim_{x\to c}\dfrac{u(x)}{v(x)}$limit, start subscript, x, \to, c, end subscript, start fraction, u, left parenthesis, x, right parenthesis, divided by, v, left parenthesis, x, right parenthesis, end fraction, si lorsque $x$x tend vers $c$c, $u(x)$u, left parenthesis, x, right parenthesis ET $v(x)$v, left parenthesis, x, right parenthesis tendent vers $0$0 ou vers $\pm\infty$plus minus, infinity. $\operatorname{}\operatorname{}\operatorname{}\operatorname{}\operatorname{}$

Selon cette règle, si la limite $\displaystyle\lim_{x\to c}\dfrac{u'(x)}{v'(x)}$limit, start subscript, x, \to, c, end subscript, start fraction, u, prime, left parenthesis, x, right parenthesis, divided by, v, prime, left parenthesis, x, right parenthesis, end fraction existe, alors :

On veut déterminer $\displaystyle\lim_{x\to 0}\dfrac{7x-\sin(x)}{x^2+\sin(3x)}$limit, start subscript, x, \to, 0, end subscript, start fraction, 7, x, minus, sine, left parenthesis, x, right parenthesis, divided by, x, squared, plus, sine, left parenthesis, 3, x, right parenthesis, end fraction.

Lorsque $x$x tend vers $0$0, le numérateur et le dénominateur de $f(x)=\dfrac{7x-\sin(x)}{x^2+\sin(3x)}$f, left parenthesis, x, right parenthesis, equals, start fraction, 7, x, minus, sine, left parenthesis, x, right parenthesis, divided by, x, squared, plus, sine, left parenthesis, 3, x, right parenthesis, end fraction tendent vers $0$0. On obtient la forme indéterminée $\dfrac{0}{0}$start fraction, 0, divided by, 0, end fraction. On va utiliser la règle de l’Hospital.

D'après ce calcul, $\displaystyle\lim_{x\to 0}\dfrac{\operatorname{}[7x-\sin(x)]'}{\operatorname{}[x^2+\sin(3x)]'}=2$limit, start subscript, x, \to, 0, end subscript, start fraction, open bracket, 7, x, minus, sine, left parenthesis, x, right parenthesis, close bracket, prime, divided by, open bracket, x, squared, plus, sine, left parenthesis, 3, x, right parenthesis, close bracket, prime, end fraction, equals, 2, donc la limite de $f(x)$f, left parenthesis, x, right parenthesis quand $x$x tend vers $0$0 est égale à $2$2.

On doit déterminer $\displaystyle\lim_{x\to 0}(1+2x)^{^{\LARGE\frac{1}{\sin(x)}}}$limit, start subscript, x, \to, 0, end subscript, left parenthesis, 1, plus, 2, x, right parenthesis, start superscript, start superscript, start fraction, 1, divided by, sine, left parenthesis, x, right parenthesis, end fraction, end superscript, end superscript. Lorsque $x$x tend vers $0$0, on obtient la forme indéterminée $1^{^{\Large+\infty}}$1, start superscript, start superscript, plus, infinity, end superscript, end superscript.

Au lieu d'étudier directement la limite de l'expression donnée, on étudie celle de son logarithme népérien. Autrement dit si $y=(1+2x)^{^{\LARGE\frac{1}{\sin(x)}}}$y, equals, left parenthesis, 1, plus, 2, x, right parenthesis, start superscript, start superscript, start fraction, 1, divided by, sine, left parenthesis, x, right parenthesis, end fraction, end superscript, end superscript, on va d'abord calculer $\displaystyle\lim_{x\to 0}\ln(y)$limit, start subscript, x, \to, 0, end subscript, natural log, left parenthesis, y, right parenthesis. et on en déduira $\displaystyle\lim_{x\to 0}y$limit, start subscript, x, \to, 0, end subscript, y

$\ln(y) =\dfrac{\ln(1+2x)}{\sin(x)}$natural log, left parenthesis, y, right parenthesis, equals, start fraction, natural log, left parenthesis, 1, plus, 2, x, right parenthesis, divided by, sine, left parenthesis, x, right parenthesis, end fraction

Lorsque $x$x tend vers $0$0, le numérateur et le dénominateur de $\dfrac{\ln(1+2x)}{\sin(x)}$start fraction, natural log, left parenthesis, 1, plus, 2, x, right parenthesis, divided by, sine, left parenthesis, x, right parenthesis, end fraction tendent vers $0$0. On obtient la forme indéterminée $\dfrac{0}{0}$start fraction, 0, divided by, 0, end fraction. Pour lever cette indétermination, on utilise la règle de L’Hospital.

$\displaystyle\lim_{x\to 0}\ln(y)=2$limit, start subscript, x, \to, 0, end subscript, natural log, left parenthesis, y, right parenthesis, equals, 2, donc $\displaystyle\lim_{x\to 0}y=e^2$limit, start subscript, x, \to, 0, end subscript, y, equals, e, squared.