Know the Limitations of Using a Calculator to Find Limits

By Mark Ryan

Although calculators have come a long way, they still can’t do everything for you in calculus class. You should think of your calculator as one of several tools at your disposal for solving limits — not as a substitute for algebraic techniques.

Many calculus problems can be done algebraically, graphically, and numerically. When possible, you should use two or three of the approaches. Each approach gives you a different perspective on a problem and enhances your grasp of the relevant concepts.

You can use the calculator methods to supplement algebraic methods, but don’t rely too much on them. First of all, the non-CAS-calculator techniques (CAS stands for ‘Computer Algebra System’) won’t allow you to deduce an exact answer unless the numbers your calculator gives you are getting close to a number you recognize — like 9.999 is close to 10, or 0.333332 is close to


or perhaps you recognize that 1.414211 is very close to


But if the answer to a limit problem is something like


you probably won’t recognize it. The number


is approximately equal to 0.288675. When you see numbers in your table close to that decimal, you won’t recognize


as the limit — unless you’re an Archimedes, a Gauss, or a Ramanujan (members of the mathematics hall of fame). However, even when you don’t recognize the exact answer in such cases, you can still learn an approximate answer, in decimal form, to the limit question.

The second calculator limitation is that it won’t work at all with some peculiar functions like


This limit equals zero, but you can’t get that result with your calculator.

By the way, even when the non-CAS-calculator methods work, these calculators can do some quirky things from time to time. For example, if you’re solving a limit problem where x approaches 3, and you put numbers in your calculator that are too close to 3 (like 3.0000000001), you can get too close to the calculator’s maximum decimal length. This can result in answers that get further from the limit answer, even as you input numbers closer and closer to the arrow-number.