## TI-89 Limits: Overview

A “limit” is the value a function “approaches” as the input or index approaches some value. In other words, it’s a theoretical point where your function maxes out. Limits are useful in calculus because some functions don’t actually reach a certain value; they come close, but not quite. An example is mixing two chemicals to form a new compound. The speed of the reaction varies, but will “stop” a certain point (at least to the naked eye). But there’s a tiny amount of chemical reaction going on, so small that you may not even be able to measure it with your instruments. There comes that point where “close is good enough,” and that is your*limit*.

With its limit command, the TI-89 makes it a snap to evaluate limits. The syntax for the function is

**limit (expression, variable, value)**. “Expression” is just the equation your want to find the limit for. “Variable” is nearly always going to be x (although check your equation!) and “value” is where you want to find the limit.

**You can access the limit() function in three different ways (which all access the same command):**

- Press F3 (Calc) and then 3 (see worked example below).
- Look in the Catalog (press l to go to the start of the l section).
- Press 2nd, then MATH. Look in the “A:Calculus” submenu.

## TI-89 Limits Example 1

**Sample problem:** **Evaluate the limit** as x approaches 8 for the following function.

f(x) = (x^{2} – 4) / (x^{2} + 6)

Step 1: Press the HOME key.

Step 2: Press the F3 button and then press 3 to select the “limit” command.

Step 3: Press Type your function into the calculator, followed by: comma x comma 8. Don’t forget to close your parentheses.

Step 4: Press ENTER .

The solution is 6/7, or .857.

Sometimes your problem might be phrased differently, as in this next example.

## TI-89 Limits Example 2

**Sample problem:** **Describe the limits** of the function f(x) = (x^{3} + 7) / (x^{2} + 3) symbolically as x approaches 0.

Step 1: Press the HOME key.

Step 2: Press F3 and then 3 to select the limit command.

Step 3: Press ( X ^ 3 + 7 ) ÷ ( X ^ 2 + 3 ) , X , 0

Step 4: Press ENTER.

The solution is 7/3 or 2.3333.

**Tips:**

- If you get the error “too few arguments,” make sure you have entered the correct amount of parentheses.
- The “X” variable and the multiplication symbol “x” look almost identical on the TI-89 keypad. Make sure you are pressing the right key — look to the equation to ensure you have typed the correct function into the calculator.

That’s how to use TI 89 Limits. You’re done!

**References:**

Clausen, C. Applications of Calculus I: Chemical Kinetics The Derivative as a Function

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