Question Use Newton's method with the specified initial approximation \( x_{0} \) to find \( x_{2} \), the third approximation to the root of the equation given below. Round your answer to the nearest thousandth. \[ f(x)=x^{3}-9 x-3, \quad x_{0}=1 \] Provide your answer below: \( x_{2} \approx \square \)

s] Using calculus and showing all working, graph the function \[ f(x)=\frac{x+2}{x^{2}-1} \]

Proporre un esempio di funzione \( g(x) \) tale che \[ \lim _{x \rightarrow+\infty} \frac{g(x)}{3 \log ^{2} x+1}=+\infty \]

Decide whether the following statement is true or false. The cube function is odd and is increasing on the interval \( (-\infty, \infty) \). Choose the correct answer below. False True

\( \lim _ { n \rightarrow + \infty } \frac { n ^ { 5 } + \sin n } { 3 ( \arctan n ) n ^ { 5 } } \)

5. [12 marks] Using calculus and showing all your working, sketch the function \[ f(x)=\frac{x}{x^{2}-4} \] noting the domain of \( f(x) \), any intercepts or asymptotes, symmetry, regions where the function is increasing or decreasing and any relative extrema. Note: You don't have to consider the second derivative and concavity.