TI 89 Calculus > Oblique Asymptotes on the TI-89

An **asymptote **is a line or a curve that approaches a curve arbitrarily closely as it tends towards infinity; the distance between the curve and the asymptote approaches zero as they head toward infinity. The type of asymptote seen most frequently is either horizontal or vertical. However, they can be nonlinear as well. When a linear asymptote is not parallel to the x- or y-axis, it is called an **oblique asymptote** (also called a nonlinear asymptote or slant asymptote).

You can find **oblique asymptotes** by long division. You can also find nonlinear asymptotes on the TI-89 graphing calculator by using the *propFrac(* command, which rewrites a rational function as a polynomial plus a proper fraction. The parts of the proper fraction give you information about the nonlinear asymptotes for the function.

## Oblique Asymptotes on the TI-89: Example 1

Sample Problem: Find the oblique asymptote for the following function:

f(x) = (x^{2} – 3x + 5) / (x + 4)

Step 1: Press the HOME key

.

Step 2: Press F2 and then 7 to select the “propFrac(” command.

Step 3: Press ( x ^ 2 – 3 x + 5 ) ÷ ( x + 4 ) ).

Step 4: Press the ENTER key.

The result is the sum of a proper fraction (33 / x + 4) and a linear polynomial function (x – 7). The linear function y = x – 7 is the equation of the oblique asymptote. You can use this method to find any oblique asymptote on the TI-89.

That’s it! You’re done!

## Find Nonlinear Asymptotes: Example 2

**Sample problem:** Find the nonlinear asymptotes for the function: f(x) = ^{(x3 – 8x2 + x + 10)}⁄_{(x – 6)}

Step 1: **Press the HOME key**.

Step 2: Press F2 and then press 7 to select the **propFrac(** command.

Step 3: **Type the function into the calculator**. To enter the function, press the following keys:( x ^ 3 – 8 x ^ 2 + x + 1 0 ) ÷ ( x – 6 ) ).

Step 4: **Press the ENTER key.**

The result is the sum of a proper fraction (^{-56}⁄_{x2 – 2x – 11}) and a quadratic function (x^{2} – 2x – 11). The quadratic function y = x^{2} – 2x – 11 is the equation of the nonlinear asymptote. You can use this method to find any nonlinear asymptote on the TI-89.

*That’s it! You’re done! *

**Tip:** Makes sure you enclose the whole equation by parentheses, otherwise you won’t get the right result for the propfrac(command.

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