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Math101learn.math101.caInfinite 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.
Fully worked example
Graphical or geometric meaning
Near $x=2$, one branch falls without bound and the other rises without bound. The graph approaches the vertical line but never needs to touch it. A table should sample both sides because magnitudes alone conceal the sign change.
Common mistakes and why they fail
Verification and reasonableness checks
- Evaluate at close points on both sides using sign-aware arithmetic.
- Confirm any vertical asymptote survives cancellation.
- Distinguish an unbounded limit from an undefined function value alone.
Read a vertical asymptote one side at a time
An infinite limit describes unbounded behavior, not a value attained by the function. At $x=a$, the two sides may have opposite signs, as for $1/(x-a)$; then the two-sided limit does not exist although $x=a$ is a vertical asymptote. Factor the denominator and use a sign chart rather than relying only on decimal samples. An even-multiplicity factor keeps its sign across the point, while an odd-multiplicity factor changes it. Cancellation also matters: a canceled factor can leave a removable hole instead of an asymptote, while the original domain exclusion remains. State each one-sided limit before drawing a two-sided conclusion.
Practice
- Find $\lim_{x\to0^+}1/x$.
- Find $\lim_{x\to0^-}1/x^2$.
- Does $\lim_{x\to0}1/x$ exist?
Answers and brief solutions
- $+\infty$.
- $+\infty$.
- No; the one-sided limits differ.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is lim(x→3−) 1/(x−3)?
- The numerator is positive.
- The denominator approaches 0 through negative values.
- The quotient decreases without bound, so the limit is −∞.
End of lesson
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