The expression in the square bracket [ (t) x (tpx) ] evaluated at t=0 is zero.
And the expression [ (t) x (tpx) ] evaluated at t=(w-x) is zero for the following reasons:
The probability that a life aged x survives for at least (w-x) years is 0
w is defined as the maximum age of an individual i.e. it is assumed the individual will not live past the age w (it is the limiting age)
(w-x) is the maximum number of years a life could still live from its current age x
- so if a life has already survived x years, the probability that it survives at least a further (w-x) years would take the individual past its limiting age (try drawing a timeline to illustrate the concept)
in other words, the probability that a life now aged x years old is still alive after (w-x) years is 0
Drawing a timeline can be quite useful when working with survival probabilities!