3) On considère l'application : \( \begin{array}{c}g:[0 ; 1]\end{array} \rightarrow+\left[\begin{array}{c}\left.0 ; \frac{1}{2}\right] \\ x\end{array}\right) \) a) Montrer que \( \left(\forall(a ; b) \in[0 ; 1]^{2}\right):(a b=1) \Rightarrow(a=b=1) \) b) Démontrer que \( \left(\forall(a ; b) \in[0 ; 1]^{2}\right): f(a)-f(b)=\frac{(a-b)(1-a b)}{\left(a^{2}+1\right)\left(b^{2}+1\right)} \) c) En déduire que \( g \) est injective.

\( \operatorname { lx } _ { x \rightarrow 0 } = \sqrt { 1 + 3 x - 1 } \quad \lim \neq \frac { 0 } { 0 } \)

4. (8 points) Calculate the following limit. Write DNE if the limit does not exist \( \lim _{x \rightarrow-1}\left(x^{2}-\frac{3 x}{x+2}\right)= \) 5. (16 points) Use the following phrases to complete the statements below to give

\( \int _{}^{}\frac{dx}{x^{2}-8x-16} d \)

7. (8 points) Calculate the following limits. Write DNE if the limit does not exist. (a) \( \lim _{x \rightarrow 0^{+}} \frac{4}{x^{3}}= \) (b) \( \lim _{x \rightarrow 0^{-}} \frac{4}{x^{3}}= \) (c) \( \lim _{x \rightarrow 0} \frac{4}{x^{3}}= \) 8. ( 20 points) Suppose \( f \) and \( g \) are functions such that

\( \operatorname { lol } _ { x \rightarrow 0 } f x = \sqrt { 1 + 3 x } - 1 \)