` Let f(t)=|{:(cos t,,t,,1),(2 sin t,,t,,2t),(sin t,,t,,t):}|` .Then find `lim_(t to 0) (f(t))/(t^(2))`

Explanation

we have
`f(t) = |{:(cos t,,t,,1),(2 sin t,,t,,2t),(sin t,,t,,t):}|`
`rArr (f(t))/(t^(2)) =(1)/(t^(2)) |{:(cos t,,t,,1),(2 sin t,,t,,2t),(sin t,,t,,t):}|`
`=|{:( cos t,,t,,1),( (2 sin t)/(t),,1,,2),((sin t)/(t),,1,,1):}|`
`"[Dividing " R_(2)" and "R_(3)" by "'t'"]"`
`rArr underset(t to 0)("lim") .(f(t))/(t^(2)) = |{:(underset(t to 0)("lim") cos t,,underset(t to 0)("lim t "),,underset(t to 0)("lim 1 ")),(underset(t to 0)("lim") . (2sin t)/(t) ,,underset(t to 0)("lim 1"),,underset( t to 0)("lim 2")),(underset( t to 0)("lim") . (sin t)/(t) ,,underset( t to 0)("lim 1"),,underset(t to 0)("lim 1 ")):}|`
`=|{:(1,,0,,1),(2,,1,,2),(1,,1,,1):}|" ""(" :' underset(t to 0)(" lim ") (sin t)/(t)=1)`
` =1(1-2)-0+1(2-1)`
`=0`