Question

Se considera functia f:R→R, f(x)= -$x^{3}$ + 3x + 2
Aratati ca$\lim_{x \to2}$ $\frac{f(x)}{x-2}$ = -9

Answer 1

$f(x)=-x^3+3x+2 \\\\ f(x)= -x^3+2x^2-2x^2+4x-x+2 \\\\ f(x)= -x^2(x-2)-2x(x-2)-(x-2) \\\\ f(x)= (x-2)(-x^2-2x-1) \\\\ f(x)=-(x-2)(x^2+2x+1) \\\\ \underline{f(x)=-(x-2)(x+1)^2} \\\\\\\\ \lim_{x \to 2} \frac{f(x)}{x-2}= \lim_{x \to 2} \frac{- (x-2)(x+1)^2}{x-2}= \lim_{x \to 2} \ \ -(x+1)^2= \\\\ =\lim_{x \to 2} \ \ -(2+1)^2 = \lim_{x \to 2} \ \ -3^2 \\\\\\ \Longrightarrow \boxed{\lim_{x \to 2} \frac{f(x)}{x-2}= -9} \ \ \ \ c.c.t.d.$