Unit 1: Limit, Continuity, and Differentiability

Limit of a Function

Epsilon-Delta (ε-δ) Definition

The formal definition of a limit is a cornerstone of calculus. It gives a precise meaning to the statement "the limit of f(x) as x approaches 'a' is L".

Definition: We say lim (x→a) f(x) = L if for any given number ε > 0 (no matter how small), there exists a corresponding number δ > 0 such that:
|f(x) - L| < ε whenever 0 < |x - a| < δ

• |f(x) - L| < ε means "the distance between f(x) and L is less than ε". This is the output tolerance.
• |x - a| < δ means "the distance between x and 'a' is less than δ". This is the input tolerance.
• The 0 < ... part just means that x ≠ a. We don't care what happens *at* x = a, only *near* it.

In simple terms: "You can make f(x) as close as you want (ε) to L, just by making x close enough (δ) to 'a'."

Cauchy's Criterion for Existence of Limit

This is a theoretical tool that lets us prove a limit *exists* without knowing what the limit's value (L) is.

Statement (without proof): A function f(x) has a finite limit as x approaches 'a' if and only if for every ε > 0, there exists a δ > 0 such that:
|f(x₁) - f(x₂)| < ε whenever 0 < |x₁ - a| < δ and 0 < |x₂ - a| < δ.

In simple terms: "If x is close to 'a', the function's values f(x) must be close to each other."

Continuity

Epsilon-Delta (ε-δ) Definition

Continuity is a more strict condition than just having a limit. A function is continuous at a point if the limit *exists*, the function is *defined* at that point, and the *limit equals the function's value*.

Definition: A function f(x) is continuous at a point x = a if for any ε > 0, there exists a δ > 0 such that:
|f(x) - f(a)| < ε whenever |x - a| < δ

Note the difference from the limit definition:

1. We are comparing f(x) to f(a), not some limit L.
2. We no longer have 0 < |x - a|. This means we *include* the point x = a itself, which is fine since |f(a) - f(a)| = 0, which is always less than ε.

Types of Discontinuities

If a function is not continuous at x = a, it is discontinuous. There are several types:

1. Removable Discontinuity:
   • The limit lim (x→a) f(x) = L exists.
   • But *either* f(a) is not defined, *or* f(a) ≠ L.
   • Example: f(x) = (x² - 1) / (x - 1) at x = 1. The limit is 2, but f(1) is undefined. It's a "hole" in the graph that can be "removed" by defining f(1) = 2.
2. Jump Discontinuity (or Discontinuity of the First Kind):
   • The left-hand limit lim (x→a⁻) f(x) and the right-hand limit lim (x→a⁺) f(x) both exist, but they are *not equal*.
   • Example: The greatest integer function f(x) = [x] at x = 2. The left limit is 1, and the right limit is 2. The graph "jumps".
3. Discontinuity of the Second Kind (or Essential Discontinuity):
   • At least one of the one-sided limits (left or right) does not exist.
   • Example (Infinite Discontinuity): f(x) = 1/x at x = 0. The limits go to +∞ and -∞.
   • Example (Oscillating Discontinuity): f(x) = sin(1/x) at x = 0. The function oscillates infinitely near 0, so no limit exists.

Differentiability

Definition of a Derivative

A function f(x) is **differentiable** at a point x = a if its derivative exists at that point. The derivative, f'(a), represents the instantaneous rate of change, or the slope of the tangent line to the graph at that point.

Definition (First Principle):
f'(a) = lim (h→0) [f(a + h) - f(a)] / h

This limit must exist (and be a finite number).

Relation between Continuity and Differentiability

This is a fundamental theorem in calculus.

Theorem: Differentiability implies Continuity.
If a function f(x) is differentiable at x = a, then it *must* be continuous at x = a.

Successive Differentiation

This is the process of differentiating a function multiple times. The result of the first derivative is the *first derivative*, differentiating that gives the *second derivative*, and so on.

• First Derivative: y' or f'(x) or dy/dx
• Second Derivative: y'' or f''(x) or d²y/dx²
• n-th Derivative: y⁽ⁿ⁾ or f⁽ⁿ⁾(x) or dⁿy/dxⁿ

n-th Derivatives of Standard Functions

Finding a pattern for the n-th derivative is a common problem.

Leibnitz's Theorem

This theorem provides a formula for the n-th derivative of a product of two functions. It is an extension of the product rule.

If y = u·v, where u and v are functions of x, then the n-th derivative of y is:

(uv)ₙ = C(n, 0)uₙv₀ + C(n, 1)uₙ₋₁v₁ + C(n, 2)uₙ₋₂v₂ + ... + C(n, r)uₙ₋ᵣvᵣ + ... + C(n, n)u₀vₙ

Where:

• uₙ and vₙ represent the n-th derivatives of u and v.
• v₀ = v and u₀ = u (the original functions).
• C(n, r) = n! / (r! (n-r)!) is the binomial coefficient ("n choose r").

Example: Find the 2nd derivative of y = x² sin(x) using Leibnitz's theorem.
Here n = 2. Let u = sin(x) and v = x² (Tip: Choose 'v' as the function that terminates to zero after a few derivatives).
u = sin(x), u₁ = cos(x), u₂ = -sin(x)
v = x², v₁ = 2x, v₂ = 2
y₂ = C(2, 0)u₂v₀ + C(2, 1)u₁v₁ + C(2, 2)u₀v₂
y₂ = (1)·(-sin(x))·(x²) + (2)·(cos(x))·(2x) + (1)·(sin(x))·(2)
y₂ = -x²sin(x) + 4x cos(x) + 2sin(x)

Unit 1: Exam Quick Tips