Unbounded behavior · one-sided limits · vertical asymptotes

Infinite Limits

Infinite limits describe function values that grow beyond every finite bound as the input approaches a point. They are statements about nearby behavior and direction of growth, not claims that the function actually takes the value infinity.

One-sided behavior first

A vertical asymptote can look completely different from its two sides.

Approaching from the left and right are separate experiments. A two-sided statement is justified only after those one-sided behaviors are compatible.

011x\frac{1}{x}
x → 0⁻
−∞-\infty
x → 0⁺
+∞+\infty

The one-sided limits have opposite signs. There is no single two-sided extended limit at zero.

021x2\frac{1}{x^2}
x → 0⁻
+∞+\infty
x → 0⁺
+∞+\infty

Both sides grow without bound in the positive direction, so the two-sided infinite-limit statement is meaningful.

One-sided asymptote lab

Approach zero from one side without ever setting x = 0.

Compare f(x)=1/x with f(x)=1/x². The first changes sign across zero; the second grows positive on both sides. Infinite-limit notation describes this unbounded trend, not a function value equal to infinity.

distance to 01.0e-1
Four distinctions

Infinity notation summarizes a trend, not a destination.

The most common errors come from treating ∞ as an ordinary number or skipping one-sided analysis near a singular point.

01

Unbounded near a point

An infinite limit describes arbitrarily large magnitude in a punctured neighborhood of the input. It does not assign the function a value of infinity.

02

One-sided sign matters

The symbols +∞ and −∞ describe direction of unbounded growth. Left and right behavior must be checked independently.

03

Vertical asymptote

The line x = a is a vertical asymptote when at least one one-sided limit as x approaches a is +∞ or −∞.

04

Undefined point

A function can be undefined at the asymptote and still have precise one-sided limit statements describing its nearby behavior.

Formal reading

“Tends to +∞” means eventually larger than every chosen bound.

For a right-hand limit such as lim⁡x→a+f(x)=+∞\lim_{x\to a^+} f(x)=+\infty, the formal statement says that for every real bound M, there is a sufficiently small right-hand neighborhood of a in which f(x) exceeds M.