Solve Limit Problems on a Calculator Using the Arrow-Number

By Mark Ryan

A limit problem asks you to determine what the y value of a function is zeroing in on as the x value approaches a particular number. You can solve a limit problem with your calculator using the arrow-number.

First, store a number into x that’s extremely close to the arrow-number, enter the limit expression in the home screen, and hit enter. If you get a result really close to a round number, that’s your answer—you’re done. If you have any doubt about the answer, just store another number into x that’s even closer to the arrow-number, get back to the limit expression, and hit enter again. This will likely give you a result even closer to the same round number—that’s it; you’ve got it.

This method can be the quickest, but it often doesn’t give you a good feel for how the y values zero in on the result. To get a better picture of this process, you can store three or four numbers into x (one after another), each a bit closer to the arrow-number, and look at the sequence of results.

Practice questions

  1. Evaluate
  2. Use your calculator to evaluate

Answers and explanations

  1. The answer is 7.
    Step 1. Use the STO button to store 6.01 into x.
    Step 2. Enter
    on the home screen and hit enter. (Note: You must be in radian mode.)
    This gives you a result of ~7.01, suggesting that the answer is 7.
    Step 3. Repeat Steps 1 and 2 with 6.001 stored into x.
    This gives you a result of ~7.001.
    Step 4. Repeat Steps 1 and 2 with 6.0001 stored into x.
    This gives you a result of ~7.0001.
    Because the results are obviously homing in on the round number of 7, that’s your answer.
  2. The answer is –11.
    You want the limit as x approaches –3, so pick a number really close to –3, like –3.0001, plug that into x in your function
    and enter that into your calculator. For example, if you’ve got a calculator like a Texas Instruments TI-84, a good way to do this is to use the STO button to store –3.0001 into x, then enter
    into the home screen and punch enter.
    The calculator’s answer is –11.0001. Because this is near the round number –11, your answer is –11. By the way, you can do this problem easily with algebra as well.