Theory: Limits and What “DNE” Means
A limit describes what values of a function approach as the input approaches a point.
We write:
\[
\lim_{x\to a} f(x) = L
\]
This means: as \(x\) gets very close to \(a\) (not necessarily equal to \(a\)), the function values \(f(x)\) get very close to \(L\).
One-sided vs two-sided limits
- Left-hand limit: \(\displaystyle \lim_{x\to a^-} f(x)\) (approach from values \(x<a\))
- Right-hand limit: \(\displaystyle \lim_{x\to a^+} f(x)\) (approach from values \(x>a\))
- Two-sided limit: \(\displaystyle \lim_{x\to a} f(x)\) exists only if the left and right limits both exist and are equal.
\[
\lim_{x\to a} f(x) \text{ exists } \Longleftrightarrow
\left(\lim_{x\to a^-} f(x) \text{ exists}\right)\ \text{and}\ \left(\lim_{x\to a^+} f(x) \text{ exists}\right)\ \text{and they are equal.}
\]
What does “DNE” mean?
DNE stands for Does Not Exist. In limits, it means the required limit value is not a single real number.
This can happen for several reasons:
1) Left and right limits are different
If
\(\displaystyle \lim_{x\to a^-} f(x) \neq \lim_{x\to a^+} f(x)\),
then the two-sided limit is DNE.
\[
f(x)=\begin{cases}
0, & x<0\\
1, & x>0
\end{cases}
\quad\Rightarrow\quad
\lim_{x\to 0^-} f(x)=0,\ \ \lim_{x\to 0^+} f(x)=1
\ \Rightarrow\ \lim_{x\to 0} f(x)\ \text{DNE}.
\]
2) The function grows without bound near \(a\)
Sometimes \(f(x)\) becomes arbitrarily large in magnitude near \(a\), for example near a vertical asymptote.
Many textbooks say “the limit is \(\pm\infty\)” (an infinite limit), but as a finite real number the limit does not exist.
\[
f(x)=\frac{1}{x^2}
\quad\Rightarrow\quad
\lim_{x\to 0} f(x)=+\infty
\quad(\text{diverges; not a finite limit}).
\]
3) Oscillation (no single value is approached)
If \(f(x)\) keeps oscillating between values and never settles near one number, the limit is DNE.
\[
f(x)=\sin\!\left(\frac{1}{x}\right)
\quad\Rightarrow\quad
\lim_{x\to 0} f(x)\ \text{DNE}.
\]
DNE vs “undefined at the point”
A very common confusion: \(f(a)\) being undefined does not automatically mean the limit is DNE.
The limit looks at values near \(a\), not necessarily at \(a\).
Example (a removable discontinuity / “hole”):
\[
f(x)=\frac{x^2-1}{x-1}
\quad(\text{undefined at }x=1)
\quad\Rightarrow\quad
\frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1\ \ (x\neq 1)
\]
\[
\lim_{x\to 1}\frac{x^2-1}{x-1}=\lim_{x\to 1}(x+1)=2.
\]
So the function value at \(x=1\) is undefined, but the limit exists and equals \(2\).
How to interpret a “DNE” result in this tool
- If the left-hand and right-hand numerical approaches stabilize to different values, the tool reports DNE for the two-sided limit.
- If values increase/decrease without bound, the tool may display \(\pm\infty\) (divergence), which is also “not a finite limit”.
- If values fluctuate instead of settling, that indicates oscillation and the tool may treat the limit as inconclusive/DNE depending on the behavior.
Tip: If the two-sided limit is DNE, try checking the left-hand and right-hand directions separately.
It’s common that one-sided limits exist even when the two-sided limit does not.
