Q. 38. (a) Find the asymptotes of the curve \[ x^{3}+2 x^{2} y-x y^{2}-2 y^{3}+3 x y+3 y^{2}+x+1 \]

If a car has an acceleration function of \( a(t) = 6t \), what is its velocity function assuming it starts from rest?

The area of a rectangular cross-section of a solid is given by the dimensions in terms of a linear function: width = \(x\) and height = \(3 - x\) for \(0 \leq x \leq 3\). Calculate the volume of the solid when these rectangles are stacked from \(x=0\) to \(x=3\).

(j.) \( y=\ln \sqrt{\frac{2}{x^{3}}} \)

(d.) \( f(x)=\left(\frac{x+5}{x^{2}+2}\right)^{2} \)

If \( \left\langle a_{n}\right\rangle \) and \( \left\langle b_{n}\right\rangle \) are two sequences such that \( \lim _{n \rightarrow \infty} a_{n}=i, \lim _{n \rightarrow \infty} b_{n}=m \), then prove that: \( \lim _{n \rightarrow \infty}\left(a_{n}+b_{n}\right)=l+m \)