Exercice 47 : Calculer les limites suivantes: \( \lim _{x \rightarrow+\infty} x(\ln (x+1)-\ln x) \) 2. \( \lim _{x \rightarrow+\infty}\left(x+\frac{\ln \left(x^{2}-x+2\right)}{x-1}\right) \) 3. \( \lim _{x \rightarrow-\infty}\left(x^{2}-1-\ln \left(1-x^{3}\right)\right) \)

Calculate the volume of the solid formed by rotating the area between \( y = \sqrt{x} \) and \( y = 0 \) from \( x = 0 \) to \( x = 1 \) about the line \( y = -1 \) utilizing the washer method.

\( \lim _ { x + \infty } [ \sqrt { x ^ { 2 } + 2 x } - ( x + 1 ) ] \)

Answer any THREE. \( \lim _{x \rightarrow 1}\left(\frac{1}{x-1}-\frac{1}{\ln x}\right) \)

3. Предел \( \lim _{x \rightarrow \infty} \frac{3 x^{2}-5 x+6}{2 x^{2}+7 x-1} \) равен...

(4) show that \( \lim _{z \rightarrow a} \) \( \frac{i z^{3}+i z-1}{2 z+3 i(z-i)} \)