Question

Найти предел функции

Answer 1

$\underset{x\rightarrow \infty}{\lim} \dfrac{(x+1)^{2} - (x+3)^{3}}{(4 - x)^{3}} = \begin{vmatrix}(x+1)^{2} \sim x^{2} \ \\ (x+2)^{3} \sim x^{3} \ \\ (4 - x)^{3} \sim -x^{3} \\ x\rightarrow \infty \ \ \ \ \ \ \ \ \ \end{vmatrix} = \underset{x\rightarrow \infty}{\lim} \dfrac{x^{2} - x^{3}}{-x^{3}} = \\\\= \underset{x\rightarrow \infty}{\lim} \bigg(\dfrac{x^{2}}{-x^{3}} - \dfrac{x^{3}}{-x^{3}} \bigg)=\underset{x\rightarrow \infty}{\lim} \bigg(1-\dfrac{1}{x} \bigg) = 1 - \dfrac{1}{\infty} = 1 - 0 = 1$