How to Evaluate a Limit with Your Calculator

By Mark Ryan

Your calculator can give you the answers to limit problems that are impossible to do algebraically. And it can solve limit problems that you could do with paper and pencil except that you’re stumped. For example, say you want to evaluate the following limit:

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The plug-and-chug method doesn’t work because plugging 5 into x produces the undefined result of

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but as with most limit problems, you can solve this one on your calculator.

Note on calculators and other technology. With every passing year, there are more and more powerful calculators and more and more resources on the Internet that can do calculus for you. These technologies can give you an answer of, for example,

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when the problem calls for an algebraic answer, or an answer of, for example,

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(not merely an approximation of 1.414), when the problem calls for a numerical answer. Older calculator models can’t give you algebraic answers, and, although they can give you exact answers to many numerical problems, they can’t give you an exact numerical answer like

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— and they also can’t give you an exact answer to the limit problem in the first example.

A calculator like the TI-Nspire (or any other calculator with CAS — Computer Algebra System) can actually do that limit problem (and all sorts of much more difficult calculus problems) and give you the exact answer. The same is true of websites like Wolfram Alpha.

Different calculus teachers have different policies on what technology they allow in their classes. Many do not allow the use of CAS calculators and comparable technologies because they basically do all the calculus work for you. So, the following discussion assumes you’re using a more basic calculator (like the TI-84) without CAS capability.

Method one for evaluating a limit using a calculator

The first calculator method is to test the limit function with two numbers: one slightly less than the arrow-number and one slightly more than it. So here’s what you do for the problem,

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If you have a calculator like a Texas Instruments TI-84, enter the first number, say 4.9999, on the home screen, press the Sto (store) button, then the x button, and then the Enter button (this stores the number into x). Then enter the function,

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and hit Enter. The result, 9.9999, is extremely close to a round number, 10, so 10 is likely your answer. Now take a number a little more than the arrow-number, like 5.0001, and repeat the process. Since the result, 10.0001, is also very close to 10, that clinches it. The answer is 10 (almost certainly). By the way, if you’re using a different calculator model, you can likely achieve the same result with the same technique or something very close to it.

Method two for evaluating a limit using a calculator

The second calculator method is to produce a table of values. Enter

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in your calculator’s graphing mode. Then go to “table set up” and enter the arrow-number, 5, as the “table start” number, and enter a small number, say 0.001, for

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— that’s the size of the x-increments in the table. Hit the Table button to produce the table. Now scroll up until you can see a couple numbers less than 5, and you should see a table of values something like the one shown here.

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Because y gets very close to 10 as x zeros in on 5 from above and below, 10 is the limit (almost certainly . . . you can’t be absolutely positive with these calculator methods, but they almost always work).

These calculator techniques are useful for a number of reasons. In addition to finding the answers to limit problems that are impossible to do algebraically, you can also use your calculator to check your answers for problems that you solve on paper. And even when you choose to solve a limit algebraically — or are required to do so — it’s a good idea to create a table like the one shown here not just to confirm your answer, but to see how the function behaves near the arrow-number. This gives you a numerical grasp on the problem, which enhances your algebraic understanding of it. If you then look at the graph of the function on your calculator, you have a third, graphical or visual way of thinking about the problem.