Theory: One-Sided Limits and Infinite Limits
A limit describes what \(f(x)\) approaches when \(x\) gets close to a target value.
In this calculator we focus on:
one-sided limits (\(x\to a^-\), \(x\to a^+\)) and limits at infinity (\(x\to \pm\infty\)),
including cases where the expression diverges to \(\infty\) or \(-\infty\), or where the limit is DNE.
1) One-sided limits
The one-sided limits are:
A two-sided limit \(\displaystyle \lim_{x\to a} f(x)\) exists exactly when both one-sided limits exist
and are equal:
\[
\lim_{x\to a} f(x)\ \text{exists}
\Longleftrightarrow
\left(\lim_{x\to a^-} f(x)\right)=\left(\lim_{x\to a^+} f(x)\right).
\]
One-sided limits are especially important when the function’s domain only exists on one side of \(a\)
(for example \(\sqrt{x}\) near \(0\)).
2) What does “\(\infty\)” mean in a limit?
Sometimes \(f(x)\) does not approach a finite number. Instead, its magnitude grows without bound.
In that case we write an infinite limit:
\[
\lim_{x\to a^+} f(x)=+\infty
\quad\text{means: } f(x)\text{ becomes arbitrarily large and positive as }x\to a^+.
\]
\[
\lim_{x\to a^-} f(x)=-\infty
\quad\text{means: } f(x)\text{ becomes arbitrarily large in magnitude and negative as }x\to a^-.
\]
In many courses, an infinite limit is described as “the limit diverges,” because it is not a finite real number.
This calculator will show \(\infty\) or \(-\infty\) when the behavior is clearly unbounded.
3) DNE in one-sided / two-sided contexts
DNE stands for Does Not Exist. For this calculator, DNE can happen for example when:
- the left and right sides approach different values (so the two-sided limit fails),
- the function is not real-defined on the required side (domain restriction),
- the values oscillate instead of settling near one value.
Classic “different sides” example:
\[
f(x)=\frac{1}{x}
\quad\Rightarrow\quad
\lim_{x\to 0^-}\frac{1}{x}=-\infty,\qquad
\lim_{x\to 0^+}\frac{1}{x}=+\infty
\]
\[
\text{So } \lim_{x\to 0}\frac{1}{x}\ \text{is DNE (the sides do not match).}
\]
4) Limits as \(x\to \pm\infty\)
Limits as \(x\to +\infty\) or \(x\to -\infty\) describe the “end behavior” of the function.
For rational functions \(\dfrac{P(x)}{Q(x)}\), a powerful rule is to compare degrees:
\[
\deg P < \deg Q \Rightarrow \lim_{x\to \pm\infty}\frac{P(x)}{Q(x)} = 0
\]
\[
\deg P = \deg Q \Rightarrow \lim_{x\to \pm\infty}\frac{P(x)}{Q(x)} =
\frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}
\]
\[
\deg P > \deg Q \Rightarrow \text{the expression grows without bound (diverges to }\pm\infty\text{).}
\]
These are exactly the rules behind horizontal asymptotes for rational functions.
Solved example (matches the Fill example)
Evaluate:
\(\displaystyle \lim_{x\to 0^+}\frac{\sqrt{x}}{x}\).
\[
\frac{\sqrt{x}}{x}=\frac{1}{\sqrt{x}}
\quad\text{(valid for }x>0\text{)}
\]
\[
x\to 0^+ \Rightarrow \sqrt{x}\to 0^+ \Rightarrow \frac{1}{\sqrt{x}}\to +\infty
\]
\[
\boxed{\displaystyle \lim_{x\to 0^+}\frac{\sqrt{x}}{x}=+\infty}
\]
Notice the domain detail: from the left (\(x\to 0^-\)), \(\sqrt{x}\) is not real-defined, so the real one-sided limit is DNE on that side.
