Question $1$: $6$ Marks

Determine the following limits:

$(1\mathrm{.}1)\underset{x-1}{lim}\frac{x^3-2}{x^{4}4x-3}$

$(1\mathrm{.}2)\underset{x3}{lim}\frac{x^3-8}{x^2-4}$

Question $2$: $22$ Marks

Differentiate the following to the appropriate independent variable and simplify where possible:

$(2\mathrm{.}1)y=\frac{1}{x}-\sqrt[\phantom{2}]{cosx}$

$(2\mathrm{.}2)y=(x^{3}2)(x^2-2x1)$

$(2\mathrm{.}3)y=\frac{\sqrt[\phantom{2}]{1ta{n}^{2}x}}{se{c}^{2}x}$

$(2\mathrm{.}4)y=ln|e^x(\frac{x-2}{x3}{)}^4|$

Question $3$: $4$ Marks

The loading distribution $y$ on a beam is given by $y=-25x^{2}45x.$

Question $4$: $9$ Marks

Given the curve $y=2x^{3}3x^2-12x.$ Find the turning points and the intercepts with the axes. Sketch the curve.