This tool searches for a large $\delta$ such that sampled inputs inside that window keep $f(x)$ within $\varepsilon$ of a proposed limit $L$. You type $f(x)$, the point $a$, the proposed limit $L$, and a positive $\varepsilon$. The page samples both sides of $a$, skips $a$ itself, and looks for the widest window that still passes, up to a cap of $10$.
The search is a numerical check, not a proof. A finite list of samples can miss a spike between them. Passing one $\varepsilon$ does not settle the definition, which asks for every positive tolerance. Read the negotiation in epsilon delta intuition before you treat a printed $\delta$ as the end of the argument.
Allowed pieces match the other numerical tools: numbers, $x$, plus, minus, times, divide, powers, parentheses, and sin cos tan sqrt abs log exp pi e. Write a times sign between a number and $x$. The text 3x is not the product 3*x.
How to read the result
The large line is the widest $\delta$ whose checked samples stayed inside the band, or a clear refusal. Under it, the note names $\varepsilon$, the cap, and the closest sampled point that escaped, when one exists. The sentence that the check is not a proof is part of every result. Do not skip it.
The plot draws $f$ near $a$. The horizontal shaded band is the strip from $L-\varepsilon$ to $L+\varepsilon$. The vertical shaded band, when a $\delta$ is found, is the input window. A red dot is a sample whose output left the strip, or a sample that was not a finite number. If the vertical band is missing, no positive $\delta$ survived the samples. The red dots are where it fails.
Read the window as a guess about size, not as a unique answer. If a $\delta$ works, every smaller positive $\delta$ works too. The page hunts a large one because that is the interesting edge. A written proof is allowed to pick a smaller convenient value and stop.
How the found delta compares with a proof
Linear Let $f(x)=3x+1$, $a=2$, and $L=7$. Then $|f(x)-7|=3|x-2|$. The outputs stay inside $\varepsilon$ exactly when $|x-2|<\varepsilon/3$. A written proof chooses $\delta=\varepsilon/3$. That choice is sharp. Any larger window contains a point that misses the band. With $\varepsilon=0.1$ the value is $0.1/3$, about $0.03333$. The finder should print a number near that. The last digits can differ by the spacing of the samples. Here the proof $\delta$ and the largest $\delta$ are the same number, because the error is a constant times $|x-a|$.
Quadratic Let $f(x)=x^{2}$, $a=3$, $L=9$, and $\varepsilon=0.1$. A standard proof first requires $|x-3|<1$, which forces $|x+3|<7$, and then chooses $\delta=\min(1,\varepsilon/7)$. For this $\varepsilon$ that is $0.1/7$, about $0.0143$. That $\delta$ is legal. It is not the largest legal $\delta$. The largest $\delta$ is the distance from $3$ to the nearest solution of $|x^{2}-9|=0.1$, which is $\sqrt{9.1}-3$, about $0.0166$. The finder is searching for that larger window. The proof only had to exhibit one $\delta$. The screen is estimating the biggest $\delta$ the samples will allow.
The same gap shows up on the reciprocal preset. For $1/x$ at $a=2$ with $L=1/2$ and $\varepsilon=0.1$, the largest $\delta$ is $1/3$, about $0.3333$. A proof that first keeps $x$ inside $(1,3)$, so the denominator stays away from $0$, can legally stop at a smaller $\delta$ such as $0.2$. Again the write up is allowed to be modest. The search is not.
Why a numerical delta never replaces the proof
The definition asks for every $x$ with $0<|x-a|<\delta$, not for a finite list. Between two samples the graph can spike, oscillate, or cross the band on a set the grid missed. Rapid oscillation, such as $\sin(1/x)$ near $0$, is the usual warning. A calm sample list is not a certificate.
The search also stops at a cap of $10$. A window that survives the cap has not been checked on the whole line. Rounding can nudge a value across the $\varepsilon$ line when the true value sits on it. And a single $\varepsilon$ is the easy half of the definition. The definition demands a rule that, given an arbitrary $\varepsilon>0$, produces a $\delta$. Passing $\varepsilon=0.1$ and failing $\varepsilon=0.01$ is a failed limit, even if the first picture looks kind.
A proof names that rule and then shows the inequality for every $x$ in the window. The picture is a reason to believe the rule is worth writing. It is not the write up. The forward paragraph still starts with "let $\varepsilon>0$", chooses $\delta$, and only then estimates $|f(x)-L|$.
Worked examples
Linear Press Linear, or leave the boxes at $3*x+1$, $a=2$, $L=7$, $\varepsilon=0.1$. The horizontal band sits around $7$. The vertical band should be about $\varepsilon/3$ wide on each side. Change $L$ to $8$ and search again. The line $3x+1$ is near $7$ at $x=2$, so the error against $8$ is about $1$, which is larger than $0.1$. No $\delta$ survives. A wrong limit fails even when the function is gentle.
Quadratic Press Quadratic. $x^{2}$ at $3$ with $L=9$ and $\varepsilon=0.1$ should return a $\delta$ near $0.0166$, a bit larger than the convenient proof value $0.1/7$. The band around $9$ is short, and the parabola is steep enough that the window is tighter than the linear example.
Removable hole Press Removable hole. The formula $(x^{2}-1)/(x-1)$ is undefined at $x=1$, and it equals $x+1$ everywhere else. With $L=2$ and $\varepsilon=0.1$, the error is $|x-1|$ on the points the definition uses, so the largest $\delta$ is $0.1$. The sampler never sits on the hole. A limit does not need a function value at $a$.
Reciprocal Press Reciprocal. $1/x$ at $a=2$ with $L=1/2$ and $\varepsilon=0.1$ should return a $\delta$ near $1/3$. The dangerous point is $x=0$, where the graph blows up. That point is $2$ units away from $a$, so a modest window never sees it. The binding side is the left, where the graph is a little steeper.
Jump Press Jump. $\mathrm{abs}(x)/x$ is $-1$ on the left of $0$ and $1$ on the right. With $L=0$ and $\varepsilon=0.5$, both sides miss the band. The page should refuse a $\delta$ and mark the escaping samples in red. No candidate $L$ would survive this $\varepsilon$, because the two outputs are $2$ apart and the band has width $1$. The long version of that argument is on the explainer.
Common mistakes
- Treating the printed $\delta$ as the only legal one. When a window works, every smaller positive window works. A proof may pick the smaller convenient value, such as $\varepsilon/7$ instead of the largest $\delta$ on the quadratic.
- Trusting one $\varepsilon$. The definition is a promise about every positive tolerance. Try a smaller $\varepsilon$ on the same function. A real limit still produces a $\delta$. A wrong $L$ stops producing one as soon as $\varepsilon$ is tighter than the gap.
- Reading a refusal as a finished proof that the limit does not exist. The samples are strong evidence of a jump, a wrong $L$, or a blowup. The write up still has to name an $\varepsilon$ that defeats every candidate, or defeat the candidate you claimed.
- Forgetting the times sign. Type
3*x+1, not 3x+1. The parser accepts arithmetic and a short list of functions. It does not accept glued coefficients.
- Expecting the point $x=a$ to be checked. The definition excludes it. A hole at $a$ is allowed. The removable preset is the picture.
Related reading
The definition, with the linear and quadratic proofs written forward, is epsilon delta intuition. The glossary lines are limit and epsilon delta definition. If you only want to see whether nearby values settle, and you do not yet have an $\varepsilon$, use the numerical limit estimator first.
The map of the first course is calculus foundations. The proof habit is real analysis and proof habits. After a limit is in place, slopes are on what a derivative actually measures and the numerical derivative calculator.