Let `f(t)=|[cost,t,1],[2sint,t,2t],[sint,t,t]|` then find `lim_(t->0) f(t)/t^2.`

Options

A. `0`
B. `-1`
C. `2`
D. `3`

Answer

A

Explanation

(a) We have
`f(t)=|(cost, t, 1),(2sin t,t,2t),(sin t,t,t)|`
Expanding along `C_(1)`,
`=cost(t^(2)-2t^(2))-2sint(t^(2)-t)+sin t(2t^(2)-t)`
`=-t^(2) cos t-(t^(2)-t)2sin t+(2t^(2)-t)sin t`
`=-t^(2) cos t-t^(2).2sin t+t.2 sin t+2t^(2)sin t`
`=-t^(2) cos t+2tsint `
`:. lim_(t to0)(f(t))/(t^(2))=lim_(t to0)((-t^(2)cost))/(t^(2))+lim_(t to0)(2t sint)/(t^(2))`
`-lim_(t to0)cost +2.lim_(t to0)(sin t)/t`
`=-+1 [ :' lim_(t to0)(sin t)/t=1` and `cos0=1]`
`=0`