00:01
We're given a vector function, and we're asked to find a limit of this vector function as t approaches a certain value.
00:11
The vector function is t squared minus t over t minus 1i plus the square root of t plus 8j plus sine of pi t over natural log of t, k, and we want to find a limit as t approaches 1 of this function.
00:30
By definition, we have that the limit as t approaches 1 exists if and only if the limits of each of the component functions at 1 exist.
00:47
So, assuming they do exist, we have this as equal to, by definition, the limit as t approaches 1 on t squared minus t over 2 minus 1, i, plus the limit as t approaches 1, on square root of t plus 8 j plus the limit as t approaches 1 of sine of pi t over natural log of t k notice that we can factor out a t from the numerator of t squared minus t over t minus 1 so this becomes a limit as t approaches 1 of and then we have a t 2 to times t minus 1 over t minus 1.
01:59
So it simply becomes the limit as t approaches 1 of t i plus.
02:10
And this is simply squared of 1 plus 8, which is square root of 9, which is 3j...