Calculer les limites suivantes :

1) `lim_{ x to -infty} x/(sqrt(x^2+1)) `

2) `lim_{ x to 0} (cosx+sinx-1)/x `

3) `lim_{ x to 1} (xsqrt(x)-1)/(x-1)`

4) `lim_{ x to (pi)/4} (1-sqrt(2)cosx)/(1-sqrt(2)sinx) `

5) `lim_{ x to -1} (x^4-x^2 +x+1)/(x+1) `

6) `lim_{ x to 0} (cosx -sqrt(cos(2x)))/(sin^2x) `

7) `lim_{ x to +infty} xsqrt(x/(x-1)) -x -1 `

4) Calculons `lim_{ x to (pi)/4} (1-sqrt(2)cosx)/(1-sqrt(2)sinx) `

On a `lim_{ x to (pi)/4} (1-sqrt(2)cosx)/(1-sqrt(2)sinx) `

` =lim_{ x to (pi)/4} (sqrt(2)(1/(sqrt(2)) -cosx))/(sqrt(2)(1/(sqrt(2))-sinx)) `

` =lim_{ x to (pi)/4} ( sqrt(2)/2 -cosx)/(sqrt(2)/2-sinx) `

` =lim_{ x to (pi)/4} ( cos((pi)/4) -cosx)/(sin((pi)/4)-sinx) `

` =lim_{ x to (pi)/4} ( cosx- cos((pi)/4))/(sinx -sin((pi)/4)) `

` =lim_{ x to (pi)/4} ( (cosx- cos((pi)/4))/(x-(pi)/4))/((sinx -sin((pi)/4))/(x-(pi)/4)) = [-(sqrt(2))/2]/[(sqrt(2))/2] = -1 `

car `lim_{ x to (pi)/4} (cosx- cos((pi)/4))/(x-(pi)/4) = cos'((pi)/4)= -sin((pi)/4)= -(sqrt(2))/2 `

et `lim_{ x to (pi)/4} (sinx- sin((pi)/4))/(x-(pi)/4) = sin'((pi)/4)= cos((pi)/4)= (sqrt(2))/2 `

`=> lim_{ x to (pi)/4} (1-sqrt(2)cosx)/(1-sqrt(2)sinx) = -1 `

5) Calculons `lim_{ x to -1} (x^4-x^2 +x+1)/(x+1) `

Méthodes

1 Pour enlever la forme indéterminée `lim_{ x to a} (f(x))/(g(x))= 0/0 ` on factorise par `x -a `

2 `P(a)= 0 <=> P(x)=(x-a)Q(x)`





On pose `P(x)= x^4-x^2+x+1 `

`=> P(-1)= 1 -1 -1+1 = 0 `

`=> P(x)= (x+1)Q(x) `

Effectuons la division euclidienne de `P` par `x+1`





`=> x^4 -x^2+x+1 = (x+1)(x^3-x^2+1) `



On a `lim_{ x to -1} (x^4-x^2 +x+1)/(x+1) `

`= lim_[ x to -1} ((x+1)(x^3-x^2+1))/(x+1) `

`= lim_{ x to -1} x^3 -x^2+1 = -1 -1 +1= -1 `

`=> lim_{ x to -1} (x^4-x^2 +x+1)/(x+1) = -1 `

Méthode 2

On pose `g(x)=x^4-x^2 +x `

`g` est dérivable sur `R` et en particulier en `
-1 `

et `g'(x)= 4x^3 -2x+1 => g'(-1)= -4+2+1= -1 `

`=> lim_{ x to -1} (x^4-x^2 +x+1)/(x+1) `

`= lim_{ x to -1} (g(x) -g(-1))/(x+1)= g'(-1)= -1 `