Question

1)Find the limit.
a:)lim t->0 (e^(-3t)i+(t^2/sin^2 t)j|+ cos 2tk)
b:)lim t->1((t^2-t)/(t-1)i+sqrt(t+ 8)j+(sin pi t)/(ln t) k)

2) . Find a vector equation and parametric equations for the line segment that joins P(-1,2,2) and Q(-3,5,1).

Answer 1

1)

a) $\lim\limits_{t\rarr{0}}(e^{-3t}\vec{i}+\frac{t^2}{sin^2t}\vec{j}+cos2t*\vec{k})=$

$=\vec{i}*\lim\limits_{t\rarr{0}}e^{-3t}+\vec{j}*\lim\limits_{t\rarr{0}}\frac{t^2}{sin^2t}+\vec{k}*\lim\limits_{t\rarr{0}}cos2t=\vec{i}*1+\vec{j}*1+\vec{k}*1=$

$=\vec{i}+\vec{j}+\vec{k}$

b)

$\lim\limits_{t\rarr{1}}({\frac{t^2-t}{t-1}}\vec{i}+\sqrt{t+8}\vec{j}+\frac{sin\pi{t}}{\ln{t}}\vec{k})=$

$=\vec{i}*\lim\limits_{t\rarr{1}}\frac{t(t-1)}{t-1}+\vec{j}*\lim\limits_{t\rarr{1}}\sqrt{t+8}+\vec{k}*\lim\limits_{t\rarr{1}}\frac{sin\pi{t}}{\ln{t}}=$

$=\vec{i}*1+\vec{j}*3-\vec{k}*\pi$

here $\lim\limits_{t\rarr{1}}\frac{\sin{\pi{t}}}{\ln{t}}=\lim\limits_{t\rarr{1}}\frac{(\sin{\pi{t}})'}{(\ln{t})'}=\lim\limits_{t\rarr{1}}\frac{\pi*{\cos{\pi{t}}}}{\frac{1}{t}}=-\pi$

2)

The vector equation of a segment is

$\vec{r}=\vec{r_{0}}+\vec{a}*t$ , here $0\le{t}\le{1}$ , $\vec{r_0}=-\vec{i}+2\vec{j}+2\vec{k}$

$\vec{a}=\vec{i}*(-3-(-1))+\vec{j}*(5-2)+\vec{k}*(1-2)=-2\vec{i}+3\vec{j}-\vec{k}$

parametric equation is

x=-1-2t, y=2+3t, z=2-t, $0\le{t}\le{1}$