Introduction
A limit describes the value that a function approaches as the input gets close to a certain point. It helps us understand the behaviour of functions near a point, especially where direct substitution does not work.
Definition of Limit
The limit of a function f(x) as x approaches a value a is written as:
lim (x → a) f(x)
This means: “What value does f(x) get close to when x gets close to a?”
Left-Hand Limit and Right-Hand Limit
Left-Hand Limit (LHL)
lim (x → a⁻) f(x)
Value approached by f(x) when x approaches a from the left side.
Right-Hand Limit (RHL)
lim (x → a⁺) f(x)
Value approached by f(x) when x approaches a from the right side.
Existence of Limit
A limit exists at x = a only if:
LHL = RHL
Basic Limit Laws
- lim (x → a) [f(x) + g(x)] = lim f(x) + lim g(x)
- lim (x → a) [f(x) − g(x)] = lim f(x) − lim g(x)
- lim (x → a) [f(x)·g(x)] = (lim f(x)) (lim g(x))
- lim (x → a) [f(x)/g(x)] = (lim f(x)) / (lim g(x)), if denominator ≠ 0
- Constant rule: lim (x → a) c = c
- Identity rule: lim (x → a) x = a
Standard Limits (Very Important)
1. lim (x → 0) (sin x / x) = 1
2. lim (x → 0) [(1 − cos x) / x²] = 1/2
3. lim (x → 0) [(tan x) / x] = 1
4. lim (x → 0) [(aˣ − 1) / x] = ln(a)
5. lim (x → 0) [(eˣ − 1) / x] = 1
Indeterminate Forms
Some limit expressions do not give direct answers. They are called indeterminate forms:
0/0, ∞/∞, 0·∞, ∞ − ∞, 0⁰, ∞⁰, 1∞
Such expressions require algebraic simplification.
Methods for Calculating Limits
1. Direct Substitution
If f(a) exists and function is continuous, plug in x = a directly.
2. Factorisation Method
Used when limit gives 0/0 form.
Example: Factor numerator/denominator and cancel common terms.
3. Rationalisation
Used for limits involving square roots.
4. Standard Limit Substitution
Rewrite expression to apply known limits like sin x / x.
Examples (Conceptual Overview)
- Evaluating limits of rational functions using factorisation
- Using standard trigonometric limits
- Checking left-hand and right-hand limits for piecewise functions
- Applying rationalisation for root-based problems
Important Results for Quick Revision
- If LHL ≠ RHL, limit does not exist.
- sin x ~ x when x → 0.
- cos x ~ 1 − x²/2 near x = 0.
- eˣ ≈ 1 + x when x → 0.
- Limits help in defining derivatives.
Practice Questions
- Find lim (x → 2) (x² − 4) / (x − 2).
- Find lim (x → 0) (sin 3x) / (x).
- Check if limit exists for f(x) = |x| at x = 0.
- Find lim (x → 0) [(√(x + 4) − 2) / x].
Simple notes for quick and clear understanding of limits.
Detailed Notes:
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