How to Do Limits on the TI-89 Titanium

Here is how to do limits on the TI-89 Titanium: one command, limit(, handles limits at a point, one-sided limits and limits at infinity. What changes is the number of arguments and how you type ∞, and both are easy to get wrong the first time.

Where limit( is and what it takes

limit( is item 3 in the F3 Calc menu. The syntax is limit(expression, variable, point), with an optional fourth argument for the direction. Command page: limit(.

F33→ limit(

Work on the HOME screen. Two settings matter: Angle should be RADIAN in MODE, because trig limits like the one below assume radians, and x should hold no stored value. If you are not sure, press 2nd F1 for Clean Up and choose NewProb before you start.

Limit at a point on the TI-89 Titanium and in Calc89
Input	Result
limit(sin(x)/x,x,0)	1

This is the classic 0/0 form. Plugging in 0 gives an undefined quotient, but the CAS recognizes the indeterminate form and returns the exact limit. You get the same behaviour for rational functions with a removable hole: type the original expression, not a simplified version, and limit( does the cancelling.

Limits at infinity

For a limit at infinity, use ∞ as the point. ∞ is the ◆ function of the CATALOG key. Negative infinity is the negation key (−) followed by ∞. The minus key next to the plus key means subtraction, so keep it for differences inside the expression.

◆CATALOG→ ∞

(−)◆CATALOG→ −∞

A typical input is limit((3x^2+1)/(x^2-4),x,∞) for a horizontal asymptote. The same syntax works for sequences: limit(expression,n,∞) tells you whether a sequence converges, which is the first step in many series tests.

Calc89 also accepts dominantTerm(, which shows the term that controls the expression as the variable grows. It is a quick way to see why a limit at infinity comes out the way it does.

Dominant term example in Calc89
Input	Result
dominantTerm(x^3+x,x,∞)	x³

One-sided limits: from the left and from the right

The fourth argument sets the direction. Any negative number means from the left, any positive number from the right. Most people use −1 and 1:

limit(f,x,a,-1) is the limit as x approaches a from the left.
limit(f,x,a,1) is the limit as x approaches a from the right.
Leaving the argument out asks for the two-sided limit.

Type the negative direction with the negation key (−), just like negative infinity. One-sided limits are what you need for piecewise functions, absolute values like abs(x)/x, and vertical asymptotes, where the function runs to ∞ on one side and to −∞ on the other.

When the answer is undef

If limit( returns undef, the two-sided limit does not exist as a single value. The usual cause is that the left and right limits differ. Run both one-sided versions to see what happens on each side; that is also what a free response question expects you to write down. An answer of ∞ or −∞ means the limit is infinite on that side, which still counts as "does not exist" in most courses.

If limit( returns the input unchanged, the CAS could not decide. Try a one-sided limit, simplify the expression first, or look at the function numerically: open the TABLE screen with ◆ F5 and step toward the point.

◆F5→ TABLE

Limits and the definition of the derivative

A common homework task is to evaluate the limit of a difference quotient. Define the function first, then type the quotient with h: Define f(x)=x^3, then limit((f(x+h)-f(x))/h,h,0). The result should agree with d(f(x),x), which is a good way to check your algebra. More on derivatives in the derivatives article.

2nd8→ d(

Common mistakes

Minus instead of negation. Before ∞ or the direction −1, use the negation key (−), not the subtraction key.
A big number instead of ∞. Plugging in 1E99 only evaluates the expression at one far away point. Use ∞ to ask for the exact limit.
A simplified expression. If you cancel a factor by hand before you type it, you may hide the hole that the question is about. Enter the original expression.
Degree mode. Trig limits are built on radians. In DEGREE mode you get a different, scaled answer.
A stored x. If x holds a number, the expression becomes a constant before the limit is taken.

In Calc89

Calc89 is a CAS graphing calculator intended for use with TI-89 Titanium, and the limit keys are the same: F3 3 for limit(, ◆ CATALOG for ∞, (−) for the direction. The results above come from verified examples of the Calc89 CAS. If a limit is printed on a worksheet, scan it with the camera; Calc89 turns it into the input line. The AI keystroke tutor then shows each key for the TI-89 Titanium, including where ∞ hides, so you can repeat the sequence on the handheld. Calc89 is free to download, Pro features as an in-app purchase.

Frequently asked questions

How do I type infinity on the TI-89 Titanium?

Press ◆ then CATALOG. For negative infinity, put the negation key (−) in front of it.

How do I do a left-hand limit?

Add a negative number as the fourth argument, for example limit(f,x,0,-1). A positive number gives the right-hand limit.

Why does limit( return undef?

The limit does not exist as a single value, usually because the left and right limits differ. Check both one-sided limits.

Does the TI-89 use L'Hôpital's rule?

The CAS handles 0/0 and ∞/∞ forms internally and returns the limit. It does not show the steps.

Limits on the TI-89: limit( at a Point, at Infinity and One-Sided