What value does a function approach, even where it has none?

Start with the hole. The values of $(x^2-1)/(x-1)$ crowd toward $2$ from both sides, yet the inspector reports $f(1)$ as undefined: the formula divides $0$ by $0$ there. Then choose Value ≠ limit: now $f(1)=3$ exists, but the nearby values still approach $2$. A limit is a statement about nearby values, never about the value at the point.

Intuition: watching from both sides

To ask for $\lim_{x\to a}f(x)$ is to ask: as $x$ gets close to $a$ — but stays different from it — do the values $f(x)$ settle near one number? Approaching from the left ($x<a$) and from the right ($x>a$) are two separate experiments. When both settle near the same number $L$, the limit is $L$. When they settle on different numbers, grow without bound, or never settle, the limit does not exist, and the way it fails is itself informative.

Definition

We write $\lim_{x\to a^-}f(x)=L$ when $f(x)$ can be made as close to $L$ as we like by taking $x<a$ close enough to $a$; similarly for $x\to a^+$. The two-sided limit exists exactly when both one-sided limits exist, are finite and are equal:

Precisely: for every tolerance $\varepsilon>0$ there is a distance $\delta>0$ such that $0<|x-a|<\delta$ implies $|f(x)-L|<\varepsilon$. The condition $0<|x-a|$ excludes $x=a$: the value $f(a)$ — if it exists — plays no part.

Mechanics: what the explorer evaluates

The table lists $f(a-\delta)$ and $f(a+\delta)$ for $\delta=1,\tfrac12,\tfrac14,\dots$ These are genuine evaluations at points different from $a$; for the rational examples they are exact. The inspector’s limit rows are not read from the table. They come from the function’s exact description: a common factor $(x-a)$ in a rational function is divided out, the sign of an unbounded side comes from the multiplicity of the root in the denominator, and the oscillation of $\sin(1/x)$ is stated analytically. The value $f(a)$ is evaluated separately and is never replaced by the limit.

Derivation: why the hole has limit 2

For $x\ne1$,

Cancelling $x-1$ is legitimate only because $x\ne1$, which is exactly the situation a limit considers. The simplified expression $x+1$ is continuous, so it approaches $1+1=2$. At $x=1$ the original expression is $0/0$, which has no value: the function has a removable hole.

For $x\sin(1/x)$ there is no cancellation, but a squeeze works: $|x\sin(1/x)|\le|x|$ for $x\ne0$, and $|x|\to0$. So the limit is $0$ even though $\sin(1/x)$ alone has no limit.

Worked example

Take $f(x)=(x^2-1)/(x-1)$ and $a=1$. With $\delta=1$: $f(0)=1$ and $f(2)=3$. With $\delta=\tfrac12$: $f(0.5)=1.5$ and $f(1.5)=2.5$. With $\delta=\tfrac14$: $1.75$ and $2.25$. Each pair sits exactly $\delta$ away from $2$, because $f(x)-2=x-1$. The left and right limits are both $2$, so $\lim_{x\to1}f(x)=2$, while $f(1)$ is undefined.

Failure cases

• Jump. For the step function, the left limit at $0$ is $-1$ and the right limit is $1$. They differ, so no two-sided limit exists.
• Unbounded. $1/x^2$ grows without bound from both sides of $0$; $1/x$ goes to $-\infty$ from the left and $+\infty$ from the right. Writing $\lim=+\infty$ records how the limit fails; $\infty$ is not a real number.
• Oscillation. $\sin(1/x)$ reaches both $-1$ and $1$ on every interval around $0$. Along $x=1/(k\pi)$ the values are $0$; along $x=2/((4k+1)\pi)$ they are $1$.

Executable reference code

The pedagogical file evaluates exact one-sided tables with Fraction, cancels the common factor and shows the jump. The NumPy file samples two sequences of $\sin(1/x)$ that approach $0$ and disagree, and checks the squeeze bound for $x\sin(1/x)$.

"""One-sided approach tables with exact Fractions (III.1). We evaluate only at x != a; f(a) is a separate question."""
from fractions import Fraction as F


def hole(x):
    if x == 1:
        raise ZeroDivisionError("(x**2 - 1)/(x - 1) is 0/0 at x = 1: undefined")
    return (x * x - 1) / (x - 1)


def approach(f, a, steps=8):
    rows = []
    for k in range(steps):
        d = F(1, 2**k)
        rows.append((d, f(a - d), f(a + d)))
    return rows


rows = approach(hole, F(1))
# f(x) - 2 = x - 1 exactly, so each sample sits exactly delta away from 2.
assert all(left == 2 - d and right == 2 + d for d, left, right in rows)

# The value at the point is not the limit: it does not even exist.
try:
    hole(F(1))
except ZeroDivisionError:
    pass
else:
    raise AssertionError("f(1) must be undefined")

# A jump: the two one-sided tables settle on different numbers.
step = lambda x: F(-1) if x < 0 else F(1)
jump = approach(step, F(0))
assert {left for _, left, _ in jump} == {-1} and {right for _, _, right in jump} == {1}

"""Two approaches to 0 that disagree for sin(1/x), and a squeeze that works for x*sin(1/x) (III.1)."""
import numpy as np

k = np.arange(1, 200)
zeros = 1 / (k * np.pi)                  # sin(1/x) = sin(k*pi) = 0
ones = 2 / ((4 * k + 1) * np.pi)         # sin(1/x) = sin(pi/2 + 2k*pi) = 1
assert np.allclose(np.sin(1 / zeros), 0, atol=1e-12)
assert np.allclose(np.sin(1 / ones), 1, atol=1e-12)
# Both sequences approach 0 from the right, yet the values approach 0 and 1: no limit exists.
assert zeros[-1] < 1e-2 and ones[-1] < 1e-2

# Squeeze: |x sin(1/x)| <= |x| -> 0.
x = np.concatenate([-np.logspace(-1, -8, 400), np.logspace(-8, -1, 400)])
assert np.all(np.abs(x * np.sin(1 / x)) <= np.abs(x))

Explore

Move $a$ away from the special point on each function: away from its trouble spot every example is continuous and its limit equals its value. On 1/x, compare the blue and green columns. On x·sin(1/x), watch the distance row shrink as the sample approaches $0$.

Common mistakes

Substituting $x=a$ answers a different question — the value, not the limit — and can produce $0/0$. Agreement of a few table entries is not a proof. A limit that is “infinite” does not exist as a real number. A two-sided limit needs both sides; one convincing side is not enough.

Numerical considerations

Floating-point evaluation of $(x^2-1)/(x-1)$ very close to $1$ suffers cancellation: both numerator and denominator lose relative accuracy. The exact rational table avoids this; a binary64 table would eventually show noise. Choosing a sequence of points also chooses what you can see: aliasing and cancellation can make a numerical table misleading in either direction.

Engineering and ML connections

Limits underlie every derivative (III.3) and integral (III.12). In computation they appear as questions of stability: does a quantity settle as a step size shrinks, a learning rate decreases or a sample size grows? Expressions such as $0/0$ appear in normalisations (dividing by a vanishing norm), and robust code handles them explicitly rather than hoping for a limiting value.

Exercises

Summary and checkpoints

A limit describes the values near $a$, from each side, not the value at $a$. It exists when both one-sided limits exist, are finite and agree. Removable holes, jumps, unbounded and oscillating behaviour are distinct ways to fail. Next, III.2 asks when the value at a point does equal its limit: continuity.