Suppose that the functions \( u \) and \( w \) are defined as follows. \[ \begin{array}{l}u(x)=2 x+2 \\ w(x)=-2 x^{2}+1\end{array} \] Find the following.

d. \( \int x^{-\frac{4}{3}} d x= \)

Exercice 27: Calculer les limites suivantes : \( \begin{array}{ll}\text { 1. } \lim _{x \rightarrow 2} \ln (2-x) & \text { 2. } \lim _{x \rightarrow 1^{+}}\left(x^{2}-1\right) \ln (x-1) \\ \text { 3. } \lim _{x \rightarrow 0^{+}}\left(x^{3}-x\right) \ln x & \text { 4. } \lim _{x \rightarrow-1^{+}}\left(\frac{1}{x+1}+\ln (1+x)\right)\end{array} \)

Prove that the sequence \( \left\langle\mathrm{S}_{n}\right\rangle \) where: \[ s_{n}=\frac{1}{1!}+\frac{1}{2!}+\ldots \ldots+\frac{1}{n!} \] is convergent.

Let \( f \) be a strictly monotonic differen- tiable function on the closed interval \( [5,10] \) such that \( f(5)=6 \) and \( f(10)= \) 26. Which of the following must be true for the function \( f \) on the interval [5,10]? I. The average rate of change of \( f \) is 4. II. The absolute maximum value of \( f \) is 26 . III. \( f(8)>0 \). (A) I only (B) II only (C) III only (D) I and II (E) I. II. and III

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. \( \lim _{x \rightarrow-\infty}\left(x^{2}-1-\ln \left(1-x^{3}\right)\right) \)