Limit of piecewise function

A student wrote in to us with this (slightly edited):

Hello Matheno!

Let me introduce myself first, I’m studying in Malaysia. I’m also student in Calculus. I need a
solution from this question given. Can you help me?

Thank you for your kindness.

Hi! Welcome to the Matheno forum, and thanks for posting this interesting problem. If you need help understanding one-sided limits, we encourage you to visit the following page:

In part i. we are asked to first calculate the limit of T(x) as x approaches 0 from values above, then interpret our answer.

  1. Calculate the limit: Since T(x) is defined at 0, and is a linear function otherwise, you can just use direct substitution to calculate the limit. In other words, you can plug x = 0 into T(x) using the top line of the piecewise function.
\lim_{x \to 0^+}T\left(x\right)=T(0)
  1. Interpret our answer: We can’t really write a statement about why this is good or bad because it asks for your opinion on the matter. However, we can maybe ask some leading questions.

    In the limit from part 1 above, “as x approaches 0” we can interpret as saying that someone’s yearly income is approaching RM0. Would it make sense to have someone pay money in taxes when we know that they don’t make any money in the first place? Does this T(x) function make people who don’t make any money pay any tax? Is this a good or a bad thing in your opinion?

In part ii. we are asked to do the same thing, but at the break-point of the piecewise function. Now, the limit could still exist at x = 10000, but only if the left and right limits are equal. So first, we must calculate each one-sided limit, and then interpret our answer again.

  1. Calculate the limit from below: For x\leq 10000, the first line of the piecewise function is true. So you can use the first line T(x) = 0.14x and direct substitution to calculate the limit from the left.

  2. Calculate the limit from above: For x\geq 10000, the second line of the piecewise function is true. So you can use the second line T(x) = 1500 + 0.21x and direct substitution to calculate the limit from the right.

  3. Decide whether the limit exists or not: If the two limits above are equal, then that number is the limit. If they are not equal, then the limit doesn’t exist.

  4. Interpret our answer: This part is particularly interesting. If the limit doesn’t exist, what would someone who makes RM9,999 pay in taxes versus someone who makes RM10,001? Does this seem fair considering that they both make essentially the same amount of money? Furthermore, how would you decide what to do with the person who makes exactly RM10000? All good questions to consider when answering this question.

To answer part iii., you first need to calculate both T(11150) and T(9995). Remember that since 11150 \gt 10000, you would use T(x) = 1500 + 0.21 to calculate T(11150), and since 9995 \lt 10000, you would use T(x) = 0.14x to calculate T(9995). When deciding which salary is better, consider their net total after taxes are taken from their salary, which can be computed by

\text{net total } = x - T(x)

Which number is higher? Would you rather take the RM11150 salary or the RM9995 salary post-taxes?

If you have any further questions about this problem, want to confirm/discuss an answer you got, or are confused by something I wrote, please let me know! Always happy to discuss.