Question

(3) (i) A broken pipe at an oil rig off the east coast of Trinidad produces a circular oil slick that is S meters thick at a distance x meters from the break. It difficult to measure the thickness of the slick directly at the source owing to excess turbulence, but for x >0 they know that s(x) = x^2/2 + 5/3x/x^3 + x^2 + 2x If the oil slick is assumed to be continuously distributed, how thick is expected to be at the source? (ii) If f(x) = {4x + 7 1< x < 2, 4x^2 - 1 2< x < 4 , determine whether the function f (x) is continuous throughout its domain? 

Answer 1

I.

$\lim\limits_{x\to0} \space \frac{\frac{x^2}{2}+\frac{5}{3}x}{x^3+x^2+2x}\\ \lim\limits_{x\to0} \space \frac{X(\frac{x}{2}+\frac{5}{3})}{X(x^2+x+2}\\ \lim\limits_{x\to0} \space \frac{\frac{x}{2}+\frac{5}{3}}{x^2+x+2}$

$\frac{\frac{x}{2}+\frac{5}{3}}{0^2+0+2}=\frac{5}{6}$

Oil slick is $\frac{5}{6}$m thick of source.

ii.

$f(x)=\begin{bmatrix} 4x+7 & 1 ≤ x ≤2\\ 4x^2-1& 2<x≤4 \end{bmatrix}$

$\lim\limits_{x\to2^-}(4x+7)\\=4(2)+7\\=13$

$\lim\limits_{x\to2^+}(4x^2-1)\\=4(2)^2-1\\=15$

$\lim\limits_{x\to2^-} f(x)\neq \lim\limits_{x\to2^+} f(x)$

f(x) is discontinuous at x=2.