A personal computer salesperson receives a base salary of $2000 per month and a commission of 5% of all sales over $10,000 during the month. If the monthly sales are $20,000 or more, then the salesperson is given an additional $500 bonus. Let $E(s)$ represent the person's earnings per month as a function of the monthly sales s. Answer parts (a) through (d).
(b) Find [tex]$\lim _{s \rightarrow 10,000} E(s)$[/tex]. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. [tex]$\lim _{s \rightarrow 10,000} E(s)=2000$[/tex]
B. The limit [tex]$\lim _{s \rightarrow 10,000} E(s)$[/tex] does not exist.
Find $E(10,000)$.
$E(10,000)=\$2000$
(c) Find [tex]$\lim _{s \rightarrow 20,000} E(s)$[/tex]. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. [tex]$\lim _{s \rightarrow 20,000} E(s)=$\square[/tex]
B. The limit [tex]$\lim _{s \rightarrow 20,000} E(s)$[/tex] does not exist.
Find $E(20,000)$.
$E(20,000)=\$\square
Answer (2)
The limit of E ( s ) as s approaches 10 , 000 is found by evaluating the left-hand and right-hand limits, both resulting in 2000 .
The value of E ( 10 , 000 ) is calculated using the appropriate piece of the function, yielding 2000 .
The limit of E ( s ) as s approaches 20 , 000 does not exist because the left-hand limit is 2500 and the right-hand limit is 3000 , which are not equal.
The value of E ( 20 , 000 ) is calculated using the appropriate piece of the function, resulting in 3000 .
Explanation
Problem Analysis We are given a problem about a salesperson's earnings, which depend on their monthly sales. The earnings function, E ( s ) , is defined piecewise based on the sales amount, s . We need to find the limit of E ( s ) as s approaches $10,000 and 20 , 000 , an d a l so f in d t h e v a l u eo f E(s)$ at these points.
Defining the Earnings Function First, let's define the earnings function E ( s ) based on the given information:
If s < 10000 , the salesperson receives only the base salary, so E ( s ) = 2000 .
If 10000 − I f s
Finding the Limit and Value at s = 10,000 Now, let's find the limit as s approaches 10 , 000 . We need to consider the left-hand limit (as s approaches 10 , 000 from below) and the right-hand limit (as s approaches 10 , 000 from above).
Left-hand limit: lim s → 1000 0 − E ( s ) = 2000 (since E ( s ) = 2000 for s < 10000 )
Right-hand limit: lim s → 1000 0 + E ( s ) = 2000 + 0.05 ( s − 10000 ) = 2000 + 0.05 ( 10000 − 10000 ) = 2000
Since the left-hand limit and the right-hand limit are equal, the limit exists and is equal to 2000 .
Therefore, lim s → 10000 E ( s ) = 2000 .
Now, let's find E ( 10000 ) . Using the definition of E ( s ) for 10000 ≤ s < 20000 , we have:
E ( 10000 ) = 2000 + 0.05 ( 10000 − 10000 ) = 2000 .
Finding the Limit and Value at s = 20,000 Next, let's find the limit as s approaches 20 , 000 . Again, we need to consider the left-hand limit and the right-hand limit.
Left-hand limit: lim s → 2000 0 − E ( s ) = 2000 + 0.05 ( s − 10000 ) = 2000 + 0.05 ( 20000 − 10000 ) = 2000 + 0.05 ( 10000 ) = 2000 + 500 = 2500
Right-hand limit: lim s → 2000 0 + E ( s ) = 2000 + 0.05 ( s − 10000 ) + 500 = 2000 + 0.05 ( 20000 − 10000 ) + 500 = 2000 + 500 + 500 = 3000
Since the left-hand limit ( 2500 ) and the right-hand limit ( 3000 ) are not equal, the limit does not exist.
Therefore, lim s → 20000 E ( s ) does not exist.
Now, let's find E ( 20000 ) . Using the definition of E ( s ) for s ≥ 20000 , we have:
E ( 20000 ) = 2000 + 0.05 ( 20000 − 10000 ) + 500 = 2000 + 0.05 ( 10000 ) + 500 = 2000 + 500 + 500 = 3000 .
Final Results In summary:
lim s → 10000 E ( s ) = 2000
E ( 10000 ) = 2000
lim s → 20000 E ( s ) does not exist.
E ( 20000 ) = 3000
Examples
Understanding limits and piecewise functions is crucial in economics and business for modeling various scenarios, such as pricing strategies or salary structures. For instance, consider a company offering a bonus to employees based on performance metrics. The bonus might kick in only after a certain threshold is reached, creating a piecewise function similar to the salesperson's earnings. Analyzing the limits at these thresholds helps the company understand the behavior of the function and make informed decisions about compensation and incentives.
The limit lim s → 10 , 000 E ( s ) is 2000 , and E ( 10 , 000 ) also equals 2000 . The limit lim s → 20 , 000 E ( s ) does not exist, while E ( 20 , 000 ) is 3000 .
;
