Connecting Limits at Infinity and Horizontal Asymptotes

Limits at infinity describe the behavior of functions as inputs approach positive or negative infinity, determining horizontal asymptotes.

Limits10–12% of exam

Understand It

Ace It

Context

What this topic is and why it exists

Limits at infinity determine the horizontal asymptotes of a function.

When x approaches positive or negative infinity, the value the function approaches is the horizontal asymptote.

For rational functions, it's about comparing the degrees of the numerator and denominator.

If the numerator's degree is higher, the function's limit is infinity or negative infinity, indicating no horizontal asymptote.

If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

If the denominator's degree is higher, the horizontal asymptote is y = 0.

The mechanism is straightforward: it's about balancing terms as x grows large.

The trap here is thinking horizontal asymptotes are boundaries the function never crosses.

They are not.

Functions can cross these asymptotes multiple times and still have them.

This distinction matters because misinterpreting it leads to incorrect conclusions about a function's end behavior.

You must identify the asymptote correctly and understand it's a trend, not a hard limit.

1 / 9