Infinite limits and vertical asymptotes

Infinity is behavior

The notation does not claim the limit exists as a real number. It records that the function grows without bound. That distinction matters when applying limit laws and interpreting graphs.

One-sided notation is essential because the two sides of a vertical asymptote may head in opposite directions.

Finding the sign on each side

Factor the denominator and use a sign chart near the suspect point. A tiny positive denominator and a positive numerator produce positive infinity; a tiny negative denominator reverses the direction.

Do not rely on a calculator snapshot, which can hide behavior behind scale or sampling.

Vertical versus horizontal asymptotes

Vertical asymptotes come from inputs approaching a finite boundary. Horizontal asymptotes describe outputs as inputs head to positive or negative infinity. A function may cross a horizontal asymptote; the statement concerns end behavior.

Worked example

Common mistakes

• Treating infinity as a number that can be substituted.
• Reporting a two-sided infinite limit when the sides disagree.
• Assuming every denominator zero creates an asymptote without checking cancellation.

Keep these ideas

• Use one-sided limits near vertical asymptotes.
• Sign analysis determines positive or negative infinity.
• Check for removable cancellation first.