Working with limits
Example
A pedestrian precinct has 80% of litter cleared each day by the council cleansing department.
This figure is achieved with the help of the tidy, responsible shoppers in the town.
However, 0.5 kg is dropped and discarded by others each day.
The precinct appears tidy if no more than 0.7 kg is scattered around.
Is it likely to appear tidy for the mayor's visit in 6 months time, or are more workers going to need to be drafted in?
Answer
80% of litter removed => 20% of litter remains, 0.5 kg dropped each day
Recurrence relation \({U_{n + 1}} = 0.2{U_n} + 0.5\)
A limit for this exists as, \(- 1\textless 0.2 \textless1\)
\(Limit = \frac{{0.5}}{{1 - 0.2}} = \frac{{0.5}}{{0.8}} = 0.625\)
The litter will not exceed 0.625 kg in the precinct and should therefore appear tidy enough for the mayor's visit.
Remember that for the limit to exist, \(- 1\textless a \textless1\) as \(n \to \infty\).
Be careful converting percentages into decimals. A mistake here will make quite a big impact on the final calculation!
\(20\% = \frac{{20}}{{100}} = 0.2\)
\(5\% = \frac{5}{{100}} = 0.05\)
\(0.5\% = \frac{{0.5}}{{100}} = 0.005\)
