Unit 1: Limits, Continuity, and Differentiability

Limit of a Function

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.

Formal (Epsilon-Delta) Definition

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', without x actually being 'a'.

Left-Hand Limit (LHL) and Right-Hand Limit (RHL)

• Left-Hand Limit (LHL): The value f(x) approaches as x approaches 'a' from the left side (values less than 'a'). lim (x → a⁻) f(x)
• Right-Hand Limit (RHL): The value f(x) approaches as x approaches 'a' from the right side (values greater than 'a'). lim (x → a⁺) f(x)

Existence of a Limit: A limit exists if and only if the Left-Hand Limit equals the Right-Hand Limit. lim (x → a) f(x) = L if and only if LHL = RHL = L

Fundamental Theorems on Limits (Algebra of Limits)

If lim (x → a) f(x) = L and lim (x → a) g(x) = M, then:

1. Sum Rule: lim (x → a) [f(x) + g(x)] = L + M
2. Difference Rule: lim (x → a) [f(x) - g(x)] = L - M
3. Product Rule: lim (x → a) [f(x) * g(x)] = L * M
4. Constant Multiple Rule: lim (x → a) [k * f(x)] = k * L (where k is a constant)
5. Quotient Rule: lim (x → a) [f(x) / g(x)] = L / M, provided M ≠ 0.
6. Power Rule: lim (x → a) [f(x)]ⁿ = Lⁿ (where n is a real number)

These rules allow us to break down complex limits into simpler ones.

Some Important Limits

These are standard results that you should memorize. They are used to solve more complex problems.

• lim (x → 0) (sin x) / x = 1 (x is in radians)
• lim (x → 0) (1 - cos x) / x = 0
• lim (x → 0) (tan x) / x = 1
• lim (x → 0) (eˣ - 1) / x = 1
• lim (x → 0) (log(1 + x)) / x = 1
• lim (x → 0) (1 + x)¹/ˣ = e
• lim (x → ∞) (1 + 1/x)ˣ = e
• lim (x → a) (xⁿ - aⁿ) / (x - a) = n * aⁿ⁻¹

Cauchy's Criterion for Existence of Limits

This is a more theoretical test that doesn't require knowing the value of the limit L.

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: As x gets closer to 'a', the values of f(x) get closer to each other.

Continuity

Continuity of a Function at a Point

A function f(x) is said to be continuous at a point x = a if it satisfies all three of the following conditions:

1. f(a) is defined (the point exists on the graph).
2. lim (x → a) f(x) exists (the LHL equals the RHL).
3. lim (x → a) f(x) = f(a) (the limit value is the same as the function's value).

If any of these conditions fail, the function is discontinuous at x = a.

Continuity on an Interval

• A function is continuous on an open interval (a, b) if it is continuous at every point in the interval.
• A function is continuous on a closed interval [a, b] if it is continuous on (a, b) AND it is "right-continuous" at 'a' and "left-continuous" at 'b'.
  • Right-continuous at 'a': lim (x → a⁺) f(x) = f(a)
  • Left-continuous at 'b': lim (x → b⁻) f(x) = f(b)

Classification of Discontinuities

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).	f(x) = (x² - 1) / (x - 1) at x = 1
Jump Discontinuity (First Kind)	LHL and RHL both exist (are finite), but are not equal.	f(x) = |x| / x at x = 0
Infinite Discontinuity (Second Kind)	At least one of the LHL or RHL is infinite (or does not exist).	f(x) = 1 / x at x = 0
Oscillating Discontinuity (Second Kind)	The function oscillates infinitely as x approaches 'a'.	f(x) = sin(1/x) at x = 0

Properties of Continuous Functions

If f(x) and g(x) are continuous at x = a, then:

• f(x) + g(x)
• f(x) - g(x)
• f(x) * g(x)
• k * f(x) (where k is a constant)
• f(x) / g(x) (provided g(a) ≠ 0)

...are also continuous at x = a.

Composition: If g(x) is continuous at x = a, and f(x) is continuous at g(a), then the composite function (f o g)(x) = f(g(x)) is continuous at x = a.

Differentiability of a Function

Differentiability at a Point

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.

• Left-Hand Derivative (LHD): f'_(a) = lim (h → 0⁻) [f(a + h) - f(a)] / h
• Right-Hand Derivative (RHD): f'+(a) = lim (h → 0⁺) [f(a + h) - f(a)] / h

A function is differentiable at 'a' if and only if LHD = RHD (and both are finite).

Geometrical Interpretation of a Derivative

The derivative f'(a) represents the slope of the tangent line to the curve y = f(x) at the point (a, f(a)).

If a function is not differentiable at a point, it means there is no unique tangent line. This can happen at:

• Corners: e.g., y = |x| at x = 0 (LHD = -1, RHD = 1)
• Cusps: e.g., y = x^(2/3) at x = 0
• Vertical Tangents: e.g., y = x^(1/3) at x = 0 (the limit is infinite)
• Discontinuities: (See next section)

Relationship between Continuity and Differentiability

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.

Chain Rule for Differentiation

The Chain Rule is used to find the derivative of a composite function.

If y = f(u) and u = g(x), then the derivative of y with respect to x is: dy/dx = dy/du * du/dx

Example: Find the derivative of y = (x² + 1)³.

1. Let u = x² + 1. Then y = u³.
2. Find the derivatives:
   • dy/du = 3u²
   • du/dx = 2x
3. Apply the Chain Rule: dy/dx = (3u²) * (2x)
4. Substitute u back in: dy/dx = 3(x² + 1)² * (2x) = 6x(x² + 1)²

Derivatives of Special Functions

Inverse Circular Functions (Inverse Trig Functions)

• d/dx (sin⁻¹ x) = 1 / √(1 - x²)
• d/dx (cos⁻¹ x) = -1 / √(1 - x²)
• d/dx (tan⁻¹ x) = 1 / (1 + x²)
• d/dx (cot⁻¹ x) = -1 / (1 + x²)
• d/dx (sec⁻¹ x) = 1 / (|x|√(x² - 1))
• d/dx (csc⁻¹ x) = -1 / (|x|√(x² - 1))

Hyperbolic Functions

• Definitions:
  • sinh x = (eˣ - e⁻ˣ) / 2
  • cosh x = (eˣ + e⁻ˣ) / 2
  • tanh x = (sinh x) / (cosh x)
• Derivatives:
  • d/dx (sinh x) = cosh x
  • d/dx (cosh x) = sinh x
  • d/dx (tanh x) = sech² x

Logarithmic Differentiation

This technique is used to differentiate functions that are:

1. A function raised to the power of another function (e.g., y = xˣ).
2. A complex product/quotient of functions (e.g., y = [f(x)g(x)] / h(x)).

Steps for y = f(x)ᵍ(ˣ)

1. Take the natural logarithm (ln) of both sides: ln y = g(x) * ln(f(x))
2. Differentiate both sides with respect to x (using implicit differentiation for ln y and the product rule on the right): (1/y) * (dy/dx) = g'(x) * ln(f(x)) + g(x) * [f'(x) / f(x)]
3. Solve for dy/dx by multiplying by y: dy/dx = y * [ g'(x) * ln(f(x)) + g(x) * f'(x) / f(x) ]
4. Substitute the original expression for y back in: dy/dx = f(x)ᵍ(ˣ) * [ g'(x) * ln(f(x)) + g(x) * f'(x) / f(x) ]

Derivatives of Implicit and Parametric Functions

Implicit Differentiation

Used when y is not explicitly defined as a function of x (e.g., x² + y² = 25).

1. Differentiate both sides of the equation with respect to x.
2. When differentiating a term with 'y', use the Chain Rule: d/dx (y²) = 2y * (dy/dx)
3. After differentiating, algebraically solve the resulting equation for dy/dx.

Example: x² + y² = 25

1. Differentiate w.r.t x: 2x + 2y * (dy/dx) = 0
2. Solve for dy/dx: 2y * (dy/dx) = -2x dy/dx = -2x / 2y = -x / y

Parametric Equations

Used when x and y are both defined as functions of a third variable, the parameter (e.g., t).

If x = f(t) and y = g(t), then:

dy/dx = (dy/dt) / (dx/dt)

Example: x = cos(t), y = sin(t) (This is a circle)

1. Find derivatives w.r.t t:
   • dx/dt = -sin(t)
   • dy/dt = cos(t)
2. Apply the rule: dy/dx = (cos t) / (-sin t) = -cot(t)