How to Find a Limit on a TI-84 with the Table Feature

The TI-84 Plus and CE do not have a limit key, so finding a limit on a TI-84 is a matter of tabulating values rather than pressing a single button. You enter the function, then build a table of x-values that close in on the point of interest from slightly below and slightly above, and read what the y-values are settling towards.

This numeric approach is the standard calculator check for a limit. It also makes the central idea visible: a limit describes where the function is heading, not necessarily what it equals at the point itself. This page shows the TBLSET and TABLE set-up, walks through a worked example whose limit exists even though the function is undefined there, and explains why the distinction matters.

Why the calculator has no limit command

A limit is a statement about all values near a point, not about any single value, and a calculator can only sample finitely many of them. Rather than pretend otherwise, the designers left the limit out and gave you the table. What the table gives you is strong numerical evidence: if the y-values from both sides close in on the same number, that number is almost certainly the limit.

The evidence is not a proof. A function can look flat in the sampled region and do something unexpected between the points, so a written answer still needs the algebra. The table confirms your reasoning; it does not replace it.

Set up TBLSET with Ask

Put the function in Y1 first, using the X,T,θ,n key for the variable. Then open the table settings with 2nd WINDOW. The screen shows the start value, the step, and two Auto/Ask switches. Set Indpnt to Ask so that you choose the x-values yourself.

Ask mode is the key step. In Auto mode the calculator steps evenly, which is fine for a rough look but useless when you need values such as 1.999 and 2.001 that never land on step boundaries. With Ask you control every x-value, so you can bracket the point precisely.

Build the table from both sides

Press 2nd GRAPH to open the TABLE screen. Type an x-value, press ENTER, and the matching Y1 appears beside it. Work in towards the target from both directions.

For a limit at x = 2, a sensible sequence runs 1.9, 1.99, 1.999 from below and 2.001, 2.01, 2.1 from above.

The left-hand column climbs towards 4 and the right-hand column falls towards 4. When both sides head for the same number, the two-sided limit exists and equals that number. If the two sides headed for different numbers, the limit would not exist — a jump, for example, behaves exactly like that.

A worked example where the limit exists but the value does not

Take f(x) = (x² − 4)/(x − 2) near x = 2. Algebra first: the numerator factorises as (x − 2)(x + 2), so for every x other than 2 the fraction simplifies to x + 2. That predicts a limit of 4 as x approaches 2.

The table agrees: 3.999 from below, 4.001 from above. The graph adds the final piece of the picture — the curve behaves exactly like the straight line y = x + 2, with one point at (2, 4) left out.

The limit and the value at the point are different things

This is the heart of the topic. The function is not defined at x = 2, because substituting 2 gives 0 ÷ 0, which the calculator rejects. Yet the limit is still 4, because a limit is about the journey towards the point, not the point itself.

Confusing the two leads to the wrong conclusion that “there is no limit because I cannot evaluate it”. The truth is the opposite: the limit is exactly what lets us talk about well-defined behaviour at a point where the formula breaks down. The failure to evaluate is a feature of the example, not an obstacle to it.

Confirm a single point with 1:value

Once the function is graphed, the CALC menu lets you test one x-value. Press 2nd TRACE for CALC, choose 1:value, and type the x you want. Entering 1.999 returns 3.999, confirming a single row of the table. Entering 2 returns an error, because the expression is undefined there — typically ERR: DOMAIN, which the DOMAIN error page explains. That message is useful evidence rather than a setback: it shows directly that the point itself is missing while the neighbourhood is well behaved.

Typing the fraction without the brackets that surround the numerator and denominator can raise ERR: SYNTAX instead, since the calculator cannot tell where the fraction ends — the SYNTAX error page covers that case. The finer points of moving around the table, such as changing the step or clearing a column, belong with our table feature tutorial.

A limit is also the idea behind the numerical tools elsewhere on this site: the gradient on our derivatives page is itself a limit, and the accumulated area on our definite integrals page is built from the same limiting process.

On the AP Calculus exams a graphing calculator such as the TI-84 Plus or Plus CE is permitted on the sections that allow one, though a calculator with a computer algebra system is prohibited. The rules change occasionally, so confirm the current wording with the AP calculator policy and your teacher.