The concept of a limit is the foundation of calculus. It describes the value that a function f(x) "approaches" as the input x "approaches" some value a.
We say that the limit of f(x) as x approaches 'a' is L, written as: lim (x → a) f(x) = L if for every number ε > 0 (epsilon), there exists a corresponding number δ > 0 (delta) such that if 0 < |x - a| < δ, then |f(x) - L| < ε.
In simple terms: You can make f(x) as close as you want (within ε) to L, just by making x close enough (within δ) to 'a'.
This is a theoretical test for the existence of a limit that does not require knowing the value (L) of the limit beforehand.
The limit lim (x → a) f(x) exists if and only if for every ε > 0, there exists a δ > 0 such that for any two points x₁ and x₂ in the deleted neighborhood of 'a' (i.e., 0 < |x₁ - a| < δ and 0 < |x₂ - a| < δ), we have: |f(x₁) - f(x₂)| < ε
In simple terms: A limit exists if, as x gets closer and closer to 'a', the corresponding values of f(x) get closer and closer to each other.
The most common problems involve checking if the Left-Hand Limit (LHL) and Right-Hand Limit (RHL) are equal.
Existence of a Limit: lim (x → a) f(x) exists if and only if LHL = RHL (and both are finite).
Example: Does the limit of f(x) = |x| / x exist at x = 0?
A function f(x) is said to be continuous at a point x = a if it satisfies all three of the following conditions:
If any of these conditions fail, the function is discontinuous at x = a.
These theorems (stated without proof as per the syllabus) describe how continuity behaves with arithmetic operations.
If f(x) and g(x) are both continuous at x = a, then:
| Type of Discontinuity | Condition | Example |
|---|---|---|
| Removable Discontinuity | lim (x → a) f(x) exists, but is not equal to f(a) (or f(a) is undefined). This is a "hole" in the graph. | f(x) = (x² - 4) / (x - 2) at x = 2. (The limit is 4, but f(2) is undefined). |
| Jump Discontinuity (First Kind) | LHL and RHL both exist (are finite), but are not equal. | f(x) = |x| / x at x = 0. (LHL = -1, RHL = 1). |
| Infinite Discontinuity (Second Kind) | At least one of the LHL or RHL is infinite (or does not exist). This is a vertical asymptote. | f(x) = 1 / x at x = 0. (LHL = -∞, RHL = +∞). |
| Oscillating Discontinuity (Second Kind) | The function oscillates infinitely as x approaches 'a', so the limit does not exist. | f(x) = sin(1/x) at x = 0. |
A function f(x) is differentiable at a point x = a if the following limit exists:
f'(a) = lim (h → 0) [f(a + h) - f(a)] / h
This limit, f'(a), is called the derivative of f(x) at x = a. It represents the instantaneous rate of change, or the slope of the tangent line to the curve y = f(x) at x = a.
A function is differentiable at 'a' if and only if LHD = RHD (and both are finite).
Theorem: Differentiability implies Continuity. If a function f(x) is differentiable at x = a, then it must be continuous at x = a.
Converse is NOT true: Continuity does NOT imply Differentiability. A function can be continuous at a point but not differentiable there.
Geometrical Reason: The graph of y = |x| has a sharp corner at x = 0, so there is no unique tangent line.
Successive differentiation means differentiating a function repeatedly.
Finding a general formula for the n-th derivative is a key skill. This is usually done by calculating the first few derivatives (f', f'', f''') and looking for a pattern.
| Function, y = f(x) | n-th Derivative, yⁿ |
|---|---|
| eᵃˣ | aⁿ * eᵃˣ |
| (ax + b)ᵐ | aⁿ * [m! / (m-n)!] * (ax + b)ᵐ⁻ⁿ |
| 1 / (ax + b) | (-1)ⁿ * n! * aⁿ * (ax + b)⁻(ⁿ⁺¹) |
| ln(ax + b) | (-1)ⁿ⁻¹ * (n-1)! * aⁿ * (ax + b)⁻ⁿ |
| sin(ax + b) | aⁿ * sin(ax + b + nπ/2) |
| cos(ax + b) | aⁿ * cos(ax + b + nπ/2) |
This theorem provides a formula for the n-th derivative of a product of two functions.
If y = u * v, where u and v are functions of x, then the n-th derivative of y is: (uv)ⁿ = ⁿC₀ uⁿv + ⁿC₁ uⁿ⁻¹v¹ + ⁿC₂ uⁿ⁻²v² + ... + ⁿCᵣ uⁿ⁻ʳvʳ + ... + ⁿCₙ uvⁿ Where:
- uⁿ is the n-th derivative of u.
- vʳ is the r-th derivative of v.
- ⁿCᵣ = n! / [r! * (n-r)!] is the binomial coefficient.
This theorem is most useful when one of the functions (say, v) is a polynomial (like x², x³). After a few derivatives, the derivatives of v will become zero, making the series finite and easy to calculate.
Example: Find the n-th derivative of y = x² * eᵃˣ.