ENGM$1280$ Problem Paper $01$

$2$

Question $3$

Evaluate the following:

$($i$)\underset{x4}{lim}\sqrt[\phantom{2}]{\frac{2x^2-7x-4}{x-4}},$

$($ii$)\underset{x0}{lim}\frac{co{s}^{2}x-1}{x},$

$($iii$)\underset{x0}{lim}\frac{(x+1{)}^2-1}{x},$

$($iv$)\underset{x\frac{}{4}}{lim}\frac{1-tanx}{sinx-cosx},$

Question $4$

The function $f$:$RR$ is defined by each of the formulae below. Using the definition of the derivative, determine in each case, the derivative $f^{\text{'}}(c)$ of $f$ at cinR.

$($i$)f(x)=x^2+sin3x,$

$($ii$)f(x)=\frac{1}{x^2+1},$

$($iii$)f(x)=x^n$ for ninN,

$($iv$)f(x)=x^{\frac{1}{3}}.$