Question

Lim x-->1( 3x² +3x -6)/ ( x²+2x-3)

Answer 1

$\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}$

Achando as raízes do numerador:

$3x^{2}+3x-6=0\\\\S=-b/a=-3/3=-1\\P=c/a=-6/3=-2$

Raízes: 2 números que quando somados dão -1 e quando multiplicados dão -2

$x'=-2\\x''=1$

Logo, podemos escrever 3x² + 3x - 6 da seguinte forma:

$3x^{2}+3x-6=3(x-[-2])(x-1)\\3x^{2}+3x-6=3(x+2)(x-1)$
_____

Achando as raízes do denominador:

$x^{2}+2x-3=0\\\\S=-b/a=-2/1=-2\\P=c/a=-3/1=-3$

Raízes: 2 números que quando somados dão -2 e quando multiplicados dão -3

$x'=1\\x''=-3$

Logo:

$x^{2}+2x-3=(x-1)(x-[-3])\\x^{2}+2x-3=(x-1)(x+3)$
______________________________

$\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\lim\limits_{x\rightarrow1}~\dfrac{3(x+2)(x-1)}{(x-1)(x+3)}\\\\\\\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\lim\limits_{x\rightarrow1}~\dfrac{3(x+2)}{x+3}\\\\\\\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\dfrac{3(1+2)}{1+3}\\\\\\\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\dfrac{3(3)}{4}\\\\\\\boxed{\boxed{\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\dfrac{9}{4}}}$
_____________________________________________

Também podemos resolver desse jeito:

$\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\dfrac{3(1)^{2}+3(1)-6}{(1)^{2}+2(1)-3}=\dfrac{0}{0}$

Quando temos limites chegando em indeterminações, podemos calculá-los com a Regra de L'hopital:

$\boxed{\boxed{\lim\limits_{x\rightarrow a}~\dfrac{f(x)}{g(x)}=\lim\limits_{x\rightarrow a}~\dfrac{f'(x)}{g'(x)}}}$

Achando a derivada do numerador:

$f(x)=3x^{2}+3x-6\\f'(x)=2\cdot3x^{2-1}+1\cdot3^{1-1}-0\\f'(x)=6x+3$

Achando a derivada do denominador:

$g(x)=x^{2}+2x-3\\g'(x)=2\cdot x^{2-1}+1\cdot2x^{1-1}-0\\g'(x)=2x+2$

Logo:

$\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\lim\limits_{x\rightarrow1}~\dfrac{6x+3}{2x+2}\\\\\\\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\dfrac{6(1)+3}{2(1)+2}\\\\\\\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\dfrac{6+3}{2+2}\\\\\\\boxed{\boxed{\lim\limits_{x\rightarrow1}~\dfrac{3x^{2}+3x-6}{x^{2}+2x-3}=\dfrac{9}{4}}}$