Math101 · Independent Ontario learning library
Math101Infinite Limits
A rigorous, example-driven guide to infinite limits, including hypotheses, method choice, verification, and practice.
The central idea
The notation $\lim_{x\to a}f(x)=+\infty$ means that $f(x)$ can be made larger than any prescribed positive bound by taking $x$ sufficiently close to $a$ (with $x\ne a$). It describes unbounded behavior, not a finite real limit. One-sided infinite limits determine the behavior on each side of a vertical asymptote.
Definitions, hypotheses, and notation
A vertical asymptote is inferred from at least one one-sided unbounded limit. Both sides need not behave alike, and the function may even be defined at the asymptote's $x$-value without changing nearby behavior. Conversely, a canceled denominator factor creates a hole rather than an asymptote when the simplified nearby expression has a finite limit.
Order of vanishing controls magnitude. Near $a$, a factor $(x-a)^2$ is positive on both sides, so its reciprocal tends to $+\infty$ from both directions. An odd power changes sign, so its reciprocal has opposite one-sided signs. Additional nonzero factors contribute only their limiting sign and scale, making factorization more informative than a calculator table alone.
Conceptual meaning
A vertical asymptote $x=a$ records input values approaching a finite number while output magnitude grows without bound. The two sides may have the same or opposite signs; that distinction is essential and is controlled by factor signs.
A dependable method and decision rule
- Factor or simplify the expression without erasing domain restrictions.
- Identify factors tending to zero and those tending to nonzero constants.
- Build a sign chart separately to the left and right of the target.
- Determine whether magnitude grows without bound and record the correct sign.
- State the two-sided conclusion only if both one-sided behaviors agree.
