Section 01

The Intuition — Approaching Without Arriving

Imagine you are walking toward a wall. With every step you halve the remaining distance. You get closer and closer — but the question a limit asks is not “did you reach the wall?” It asks: “what value are you approaching?”

This distinction is the entire heart of calculus. A limit describes the behaviour of a function near a point — not necessarily at the point itself. The function might not even be defined at that exact point, yet the limit can still exist perfectly clearly.

A concrete example — the speed of a car

At exactly the instant t = 2 seconds, what is a car’s speed? Speed is distance divided by time — but over exactly zero seconds, that is 0/0, which is undefined. Yet the car clearly has a speed. Limits solve this: we measure the average speed over smaller and smaller time intervals approaching zero, and ask what value that average approaches. That approaching value is the instantaneous speed — and it is a limit.

Section 02

The Formal Definition of a Limit

We write a limit like this:

This means: for every number ε > 0 (no matter how tiny), there exists a δ > 0 such that whenever 0 < |x − a| < δ, we have |f(x) − L| < ε. In plain English: we can make f(x) as close to L as we like, by making x close enough to a.

Worked Example — a simple limit

Find: lim (x² − 1) / (x − 1) as x → 1

Plugging in x = 1 directly gives 0/0 — undefined. But factor the numerator: (x² − 1) = (x − 1)(x + 1). Cancel the (x − 1) terms:

The function (x² − 1)/(x − 1) is undefined at x = 1 — there is a hole in the graph. But as x approaches 1 from either side, the function approaches 2. The limit exists and equals 2, even though f(1) does not exist.

Section 03

Interactive Limit Table Explorer

Choose a function and a target value. The table shows f(x) for values of x approaching the target from both sides — getting closer and closer. Watch the values converge to the limit.

Section 04

Interactive Limit Graph Visualiser

See the same functions plotted. The vertical dashed line marks the target x value. The horizontal dashed line marks the limit L. Notice the function approaches L as it nears the dashed line — even if there is a hole at that exact point.

Section 05

One-Sided Limits — Left and Right

Sometimes a function approaches different values depending on which side of a you come from. We call these one-sided limits:

Example — the jump function f(x) = |x|/x

This function equals −1 for all negative x and +1 for all positive x. As x → 0⁻ (from the left), f(x) → −1. As x → 0⁺ (from the right), f(x) → +1. The two sides disagree — so the two-sided limit does not exist.

Section 06

Limit Laws — The Arithmetic of Limits

Once you know individual limits, you can combine them using the Limit Laws — just like rules of arithmetic, but for limits. Assume lim f(x) = L and lim g(x) = M as x → a.

Section 07

Evaluation Techniques — 4 Methods

Worked Example — Rationalisation

Find: lim (√x − 2) / (x − 4) as x → 4

Direct substitution gives 0/0. Multiply by the conjugate (√x + 2)/(√x + 2):

Section 08

Special Limits — The Ones Every Calculus Student Needs

These four limits appear constantly in calculus. Memorise them — they are used in derivative proofs, Taylor series, and differential equations.

Section 09

Limits and Continuity — The Connection

A function f is continuous at x = a if three conditions all hold simultaneously:

Section 10

Limits at Infinity

What happens to f(x) as x grows without bound? This is a limit at infinity, written lim f(x) as x → ∞. The value L (if it exists) is called a horizontal asymptote of the graph.

Section 11

Real-World Uses of Limits

Section 12

Python Code

Computing limits numerically

import numpy as np

def numerical_limit(f, a, side='both', steps=10):
    """Estimate limit of f as x → a by computing f near a."""
    results = []
    for i in range(1, steps+1):
        h = 10**(-i)   # 0.1, 0.01, 0.001, ...
        if side in ('both', 'right'):
            results.append((f'x={a+h:.{i}f}', f(a + h)))
        if side in ('both', 'left'):
            results.append((f'x={a-h:.{i}f}', f(a - h)))
    return results

# Example 1: lim (x²−1)/(x−1) as x → 1
f1 = lambda x: (x**2 - 1) / (x - 1)
for label, val in numerical_limit(f1, 1):
    print(f"{label:12} → f(x) = {val:.8f}")
# All values approach 2.0

# Example 2: lim sin(x)/x as x → 0
f2 = lambda x: np.sin(x) / x
for label, val in numerical_limit(f2, 0, side='right'):
    print(f"{label:14} → {val:.10f}")
# All values approach 1.0

Symbolic limits with SymPy

from sympy import *

x = Symbol('x')

# lim (x²−1)/(x−1) as x → 1
expr1 = (x**2 - 1) / (x - 1)
print(limit(expr1, x, 1))         # 2

# lim sin(x)/x as x → 0
print(limit(sin(x)/x, x, 0))     # 1

# lim (1 + 1/x)^x as x → ∞  (definition of e)
print(limit((1 + 1/x)**x, x, oo)) # E  (Euler's number)

# lim (√x−2)/(x−4) as x → 4
print(limit((sqrt(x)-2)/(x-4), x, 4))  # 1/4

# One-sided limits
print(limit(Abs(x)/x, x, 0, '+'))  # 1  (right)
print(limit(Abs(x)/x, x, 0, '-'))  # -1 (left)

# Limit at infinity
print(limit((3*x**2 + 2*x) / (x**2 + 5), x, oo))  # 3

Section 13

Knowledge Quiz

Six questions to test your understanding of limits.