Textbook
A real function $f$ is defined in a punctured neighborhood of $a$: at all sufficiently close inputs, possibly excluding $a$ itself. The statement $\lim_{x\to a}f(x)=L$ means that for every $\varepsilon>0$ there is a $\delta>0$ such that $0<|x-a|<\delta$ implies $|f(x)-L|<\varepsilon$.
The order matters. For any required output accuracy $\varepsilon$, there must be an input tolerance $\delta$ that guarantees it. The implication must hold for all suitable nearby inputs on both sides, not just a few selected samples.
The value $f(a)$ is absent from this definition. For $f(x)=(x^2-1)/(x-1)$, the denominator vanishes at 1, but for $x\ne1$ the function equals $x+1$. Its limit as $x\to1$ is 2: the choice $\delta=\varepsilon$ works, since $|f(x)-2|=|x-1|$.
Agreement on a numerical grid does not prove a limit exists. The definition controls every sufficiently close input. Different left and right limits prevent a two-sided limit.
The choice of $\delta$ may depend on $\varepsilon$ and the point $a$, but not on the particular input within the chosen neighborhood. Once a neighborhood works, every smaller one works too. The definition therefore requires more than one fortunate sequence of inputs: sufficiently close to $a$, every permitted output must stay within the specified corridor around $L$.
Definitions and notation: Basic Analysis I.
Related concepts: Real Intervals: Are the Endpoints Included?.
In plain language
A function limit resembles a promise to fit all nearby outputs into a corridor around one number. Every requested accuracy must have a suitable neighborhood of inputs around the point under consideration. The point itself may be closed for repairs; the limit describes approaching the door rather than service inside. Convenient neighbors cannot be selected apart from inconvenient ones: the condition must hold for every allowed input sufficiently close to the point, excluding the point itself.
Example
In the constructed demonstration, inputs approach 1 from both sides for $f(x)=(x^2-1)/(x-1)$. The table illustrates the limit; the identity $|f(x)-2|=|x-1|$ above supplies the proof.
Function limits: making “approaches” precise
Further away
Inputs 0 and 2 give values 1 and 3.
Step 1 of 4: Further away
Read all steps
Further away
Inputs 0 and 2 give values 1 and 3.
- Inputs from left and right
- 02
- Values
- 1current3current
Closer
At distance 0.1 from 1, output error is also 0.1.
- Inputs
- 0.91.1
- Values
- 1.9current2.1current
Closer still
Shrinking input distance shrinks output error on both sides.
- Inputs
- 0.991.01
- Values
- 1.99included2.01included
The point itself
At input 1 the original fraction is undefined; this does not change its limit.
- Input; value; limit
- 1undefinedexcluded2included
The exact check concerns a whole neighborhood rather than one table entry. For accuracy $\varepsilon=0.01$, the choice $\delta=0.01$ works. Every input with $0<|x-1|<0.01$ then has output error below one hundredth, because that error equals its distance from 1. The entries 0.99 and 1.01 lie on this neighborhood’s boundary: the strict inequality requires closer inputs. This separates an illustration from checking the exact definition.
A limit requires a guarantee about nearby values. Substituting the point itself answers a different question.