4. Sabiendo que \( \int_{0}^{6} f(x) d x=12 ; \int_{0}^{2} f(x) d x=2 \) y \( \int_{3}^{6} f(x) d x=8 \) calcular \( \int_{0}^{2} 4 f(x) d x-\int_{2}^{3} f(x) d \)

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

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

(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 \( f(x)=\frac{2 x}{\sqrt{x^{2}+2}} \).

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

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 ... \( \begin{array}{llll}\text { a) } y=3 x^{2}-4 & \text { b) } y=3 x^{2}-11 x+7 & \text { c) } y=3 x^{2}-4 x & \text { d) } y=3 x^{2}-x-3\end{array} \)