Part (a)
We'll use the second piece of the piecewise function.
The leading terms in the numerator and denominator are 500t and t respectively. Their ratio is (500t)/(t) = 500.
As t heads to infinity, the fraction (500t-1500)/(t-1) approaches 500.
Therefore, the limiting value is 500.
It means the amount of money raised approaches $500 as time continues to march on forever. We'll never actually reach 500 itself. Rather we just get closer and closer.
============================================
Part (b)
Plug t = 4 into the first piece
f(t) = 5*2^(t+1)-3
f(4) = 5*2^(4+1)-3
f(4) = 157
Do the same for the second piece
g(t) = (500t-1500)/(t-1)
g(4) = (500*4-1500)/(4-1)
g(4) = 166.6667 approximately
Because f(4) = g(4) is not a true statement, this would mean F(t) is not continuous at t = 4. We have a jump discontinuity.
Use graphing software such as Desmos or GeoGebra to confirm this.
============================================
Part (c)
Replace M(t) with 100 and solve for t.
M(t) = 210(2^t-1)/(2^t+5)
100 = 210(2^t-1)/(2^t+5)
100(2^t+5) = 210(2^t-1)
100(w+5) = 210(w-1) ............... let w = 2^t
100w+500 = 210w-210
100w-210w = -210-500
-110w = -710
w = -710/(-110)
w = 6.45454545 approximately
2^t = 6.45454545
log(2^t) = log(6.45454545)
t*log(2) = log(6.45454545)
t = log(6.45454545)/log(2)
t = 2.6903154998514 approximately
It takes about 2.6903 days to reach $100
This value of t is in the interval 0 ≤ t ≤ 4, which makes it a valid solution.