\begin{tabular}{l} Utilice el teorema de Green para determinar el valor de la integral de línea \\ \( \oint_{c} x y^{2} d x+x^{3} d y \) \\ Donde \( C \) es el triángulo con vértices en \( (0,0),(2,0,(2,3) y(0,3) \) \\ 06 \\ \hline 8 \\ \hline 8\end{tabular}

(1) Find the value of: \( \quad \lim _{x \rightarrow 4} \frac{x^{3}-15 x-4}{x-4} \)

nction définie par \( f(x)=\frac{L}{x^{2}+1} \)

3. \( \lim _{x \rightarrow \infty} \frac{x-1}{2 x+5} \)

If the slope of the tangent to the curve: \( y=f(x) \) at any point on it equals \( 6 x+a \), where \( a \) is constant if the equation of the tangent to the curve at the point \( (1,-1) \) is \( y=4- \) \( 5 x \), then the equation of the curve is ... \( \begin{array}{llll}\text { a) } y=3 x^{2}-4 & \text { b) } y=3 x^{2}-11 x+7 & \text { c) } y=3 x^{2}-4 x & \text { d) } y=3 x^{2}-x-3\end{array} \)

(a) Determine whether \( f(x)=\frac{|x-4|-2}{x} \) is continuous at \( x=4 \). [6 marks] (b) Given \( f(x)=\frac{2 x}{\sqrt{x^{2}+2}} \). By finding the limit at appropriate values, obtain the horizontal and vertical asymptotes, if they exist. Hence, sketch the graphs of \( f(x)=\frac{2 x}{\sqrt{x^{2}+2}} \).