Need help with math questions ( student number last digit: 6, let S be "6")

Question 4.] (5.5 marks} Let S be the last nonzero digit of your student number. Let f($) be a piecewise function dened as follows e+1 mDo :l:>Do Ion PDD 1 Use these properties to calculate lim 111 (1 + ) . n - \\ .u/ (d) In this question you will be introduced to L'Hopital's Rule, further in the unit we will touch again on this interesting topic, and expand it further (note you do not need to show this statement). Suppose we have two functions x) and 9(3) such that 1:111}J f[:L') = 111% g(:s) = G. Then we have mb mh (HI) -f()) @ = a: 0 9(3) (9(1) - am) ' :1: 0 Taking limit as :1: > II] we get aw) H0) . HI) ) mb 9(3) mb (9(3) _ fan) lim 0 9(1) _ \") 91(0) :1: 3} :1: G This is known as L'Hopital's rule. 111(1+ 1:)- Use L'Hopital's rule to calculate lim 314] :1\": . . . . 1 . (e) Use the prev1ous part and the substltutlon I = i to calculate 11111 $111 (1 + ) . Note that under this I mmo substitution, when I 3- 00 we have y + I}. 1 n 1 n (3 Use that fact that ex is continuous and (1 + ) = e\"ln(1+%) to calculate lim (1 + ) . Hence n nmo '1'], write down the amount of money at the end of the year if intemst is paid continuously. Question 4.3 (7.5 marks) Suppose a lake is modelled by the region y a: > It} in the plane. The shore is the straight line 1; :1: = 1}. Suppose a boat travels on the lake with position vector phi) = ti + (1 + t2)j at time t 2 U. (a) Find the parallel and perpendicular vector projections of p(t) onto the vector 3 + j. (b) Find the distance from the boat to the shore at time t. (c) Find rate of change of the distance from the boat to the shore. then nd the range of time when the boat is moving towards shore