Approximating Limits on a Graphing Calculator

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Using a graphing calculator can make it a much faster and easier process to find the limit of a function, especially if the function is complex. This lesson will review two ways to approximate limits using a Texas Instruments TI-84.

Approximating Limits on a Graphing Calculator (TI-84)

Approximating limits by hand is a great skill to have, but it can sometimes be needlessly tedious or difficult. The Texas Instruments TI-84 comes in handy either to find limits or to check work done by hand. This can be helpful both in practice and on exams.

Although there is no program which directly approximates limits on the TI-84, there are two quick and easy methods to approximate them yourself:

  1. Limits via the TRACE command
  2. Limits via the TABLE command

In this case, we will seek to find the limit as x approaches 2 of (x+1 / x-1) or lim→2 (x+1 / x-1).

Before we make use of either of these methods to approximate lim→2 (x+1 / x-1), it is important that we set up the MODE and WINDOW within the calculator. These two commands set the calculator to deal with this type of function.


MODE determines how the calculator will function. It determines whether the calculator is in normal, scientific, or engineering mode; how many decimal places the calculator will round to; and other important parameters. Ensure that the mode is set as follows:



WINDOW will set the dimensions of our Cartesian coordinate system -- that is, how much of the graph we see. In our example, a useful window is: Xmin = -5, Xmax = 5, Xscl = 1, Ymin = -10, Ymax = 10, Yscl = 1 and Xres = 1. The mins and maxes are the bounds of the graph, while Xscl and Yscl determine what tick marks along the x and y axes represent. Xres is a variable we can just take to be 1.


Limits via the TRACE command

The first order of business is to place the equation for y = (x+1 / x-1) into our Y = :


Next, we GRAPH the function:


After the function is displayed graphically, we press the TRACE command. We are using this command to input x values close to 2, as we seek to find what the limit of (x+1 / x-1) is as x approaches 2. We pick the following x values -- 2.1, 2.01, and 2.001 -- which get incrementally closer to x = 2 from right:

x = 2.1 gives us y = 2.81


x = 2.01 gives us y = 2.98


And x = 2.001 gives us y = 2.99


We also choose some x values which get incrementally closer to x = 2 from the left: x = 1.9, 1.99, and 1.999.

x = 1.9 gives us y = 3.22


x = 1.99 gives us y = 3.02


And x = 1.999 gives us y = 3.00


From this data, we can clearly deduce that, as x approaches 2, y approaches 3; therefore, the lim→2 (x+1 / x-1) = 3

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