IB Mathematics AA AHL Informal ideas of limit, continuity and convergence Study Notes - New Syllabus
IB Mathematics AA AHL Informal ideas of limit, continuity and convergence Study Notes
LEARNING OBJECTIVE
- Continuity and differentiability
Key Concepts:
- Limits
- Continuity and differentiability
- Higher Derivatives
- IBDP Maths AA SL- IB Style Practice Questions with Answer-Topic Wise-Paper 1
- IBDP Maths AA SL- IB Style Practice Questions with Answer-Topic Wise-Paper 2
- IB DP Maths AA HL- IB Style Practice Questions with Answer-Topic Wise-Paper 1
- IB DP Maths AA HL- IB Style Practice Questions with Answer-Topic Wise-Paper 2
- IB DP Maths AA HL- IB Style Practice Questions with Answer-Topic Wise-Paper 3
Continuity and Differentiability at a Point (Informal Understanding)
Continuity at a Point
A function \( f(x) \) is continuous at a point \( x = a \) if:
- \( f(a) \) is defined.
- Limit \( \lim_{x \to a} f(x) \) exists.
- \( \lim_{x \to a} f(x) = f(a) \)
This means that the graph of the function does not have a break, hole, or jump at \( x = a \). The function values smoothly connect at that point.
Common reasons for discontinuity:
- Hole (removable discontinuity): limit exists but \( f(a) \) is not defined or not equal to the limit.
- Jump discontinuity: left-hand limit \( \ne \) right-hand limit.
- Infinite discontinuity: limit tends to infinity.
DEFINITION
If $f(x)$ is discontinuous at $a$, then
1. $f$ has a removable discontinuity at $a$ if $\lim\limits_{x \to a} f(x)$ exists.
(Note: When we state that $\lim\limits_{x \to a} f(x)$ exists, we mean that $\lim\limits_{x \to a} f(x) = L, \text{ where } L \text{ is a real number.}$)
2. $f$ has a jump discontinuity at $a$ if $\lim\limits_{x \to a^-} f(x)$ and $\lim\limits_{x \to a^+} f(x)$ both exist, but $\lim\limits_{x \to a^-} f(x) \ne \lim\limits_{x \to a^+} f(x)$.
(Note: When we state that $\lim\limits_{x \to a^-} f(x)$ and $\lim\limits_{x \to a^+} f(x)$ both exist, we mean that both are real-valued and that neither take on the values $\pm\infty$.)
3. $f$ has an infinite discontinuity at $a$ if $\lim\limits_{x \to a^-} f(x) = \pm\infty$ or $\lim\limits_{x \to a^+} f(x) = \pm\infty$.
Differentiability at a Point
A function is differentiable at a point \( x = a \) if it has a defined derivative at that point. That is:
$ \lim_{h \to 0} \frac{f(a+h) – f(a)}{h}$ exists and is finite.
Important: If a function is differentiable at \( x = a \), then it must also be continuous at \( x = a \). But the reverse is not always true — a function can be continuous at a point but not differentiable.
Informal Indicators of Non-Differentiability:
- Sharp corners or cusps (e.g., \( |x| \) at \( x = 0 \))
- Vertical tangent lines (e.g., \( \sqrt[3]{x} \) at \( x = 0 \))
- Discontinuity – if a function is not continuous at \( x = a \), it cannot be differentiable there.
Example
Is the function \( f(x) = |x| \) continuous and differentiable at \( x = 0 \)?
▶️ Answer/Explanation
Continuity:
\( f(0) = |0| = 0 \) → Defined
\( \lim_{x \to 0^-} |x| = 0 \) and \( \lim_{x \to 0^+} |x| = 0 \)
Since both one-sided limits and the function value at 0 agree, the function is continuous at \( x = 0 \).
Differentiability:
Left-hand derivative: \( \lim_{h \to 0^-} \frac{|0 + h| – |0|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1 \)
Right-hand derivative: \( \lim_{h \to 0^+} \frac{|0 + h| – |0|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1 \)
Since LHD \( \ne \) RHD, the derivative does not exist at \( x = 0 \)
Conclusion: The function \( f(x) = |x| \) is continuous but not differentiable at \( x = 0 \).
Understanding of Limits: Convergence and Divergence
Understanding of Limits
A limit describes the behavior of a function or sequence as the input approaches a particular value (including infinity). Limits help us understand how functions behave near specific points or as values grow very large.
Convergent Limits
A limit is said to converge if it approaches a finite, real number as the variable approaches a certain value.
$ \lim_{x \to a} f(x) = L \quad \text{(L is finite)} $
This means as \( x \) gets closer to \( a \), the function values get closer to \( L \).
Examples of Converging Limits:
- \( \lim_{x \to 2} (3x + 1) = 7 \)
- \( \lim_{x \to 0} \sin x = 0 \)
Divergent Limits
A limit diverges if it does not approach a finite real number. This can happen in two main ways:
- Goes to infinity or negative infinity: The function increases or decreases without bound.
Example: \( \lim_{x \to 0^+} \frac{1}{x} = \infty \) - Oscillates without approaching a single value: The function keeps changing and doesn’t settle.
Example: \( \lim_{x \to 0} \sin\left(\frac{1}{x}\right) \) does not exist (oscillates infinitely).
Left and Right-Hand Limits
We may need to evaluate limits from either side:
- \( \lim_{x \to a^-} f(x) \): from the left side of \( a \)
- \( \lim_{x \to a^+} f(x) \): from the right side of \( a \)
If both one-sided limits are equal and finite, the two-sided limit exists and is equal to that common value.
Importance of Limits
Limits form the foundation of calculus. They are essential for:
- Defining derivatives (instantaneous rate of change)
- Defining integrals (area under curves)
- Analyzing function behavior at discontinuities or boundaries
Example
Determine if the limit \( \lim_{x \to 0^+} \frac{1}{x} \) converges or diverges.
▶️ Answer/Explanation
As \( x \to 0^+ \), the denominator becomes very small positive.
Therefore, \( \frac{1}{x} \to \infty \).
Conclusion: The limit does not converge. It diverges to infinity.
Limits and Infinite Geometric Sequences
Limits and Infinite Geometric Sequences
In the context of sequences and series, limits help us understand the long-term behavior of a sequence or series as the number of terms approaches infinity.
Infinite Geometric Sequence
An infinite geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a constant ratio \( r \).
$ a, ar, ar^2, ar^3, \dots $
If \( |r| < 1 \), the terms get smaller and smaller — they converge to zero as the number of terms increases.
Infinite Geometric Series
The sum of an infinite geometric sequence is called an infinite geometric series:
$ S = a + ar + ar^2 + ar^3 + \dots $
If \( |r| < 1 \), the series has a finite limit and the sum is:
$ S_\infty = \frac{a}{1 – r} $
This formula comes from the limit of the partial sums:
$ \lim_{n \to \infty} S_n = \lim_{n \to \infty} \left( \frac{a(1 – r^n)}{1 – r} \right) = \frac{a}{1 – r} \quad \text{if } |r| < 1 $
Why? Because \( \lim_{n \to \infty} r^n = 0 \) when \( |r| < 1 \). This is a converging limit.
Divergent Case
If \( |r| \geq 1 \), then \( \lim_{n \to \infty} r^n \) does not go to zero, and the series diverges (it grows without bound or oscillates).
For example:
- \( r = 2 \): \( a + 2a + 4a + \dots \) → diverges
- \( r = -1 \): \( a – a + a – a + \dots \) → does not settle → diverges
Example
Determine if the infinite geometric series \( 3 + 1.5 + 0.75 + \dots \) converges, and find its sum.
▶️ Answer/Explanation
This is a geometric series with:
First term \( a = 3 \)
Common ratio \( r = \frac{1.5}{3} = 0.5 \)
Since \( |r| = 0.5 < 1 \), the series converges.
Use the formula:
$ S_\infty = \frac{a}{1 – r} = \frac{3}{1 – 0.5} = \frac{3}{0.5} = 6$
Final Answer: The series converges and the sum is 6.
Derivative from First Principles
Derivative from First Principles
The derivative of a function \( f(x) \) at a point \( x \) represents the instantaneous rate of change or the slope of the tangent to the curve at that point.
Definition (First Principles)
The derivative of \( f(x) \), written as \( f'(x) \), is defined using the limit:
$ f'(x) = \lim_{h \to 0} \frac{f(x + h) – f(x)}{h} $
This is called the definition from first principles or the difference quotient. It finds the gradient of the secant line between two points on the graph and then takes the limit as they get infinitely close.
Use for Polynomials
The IB syllabus restricts the application of this definition to polynomials only, such as:
- Linear functions: \( f(x) = ax + b \)
- Quadratic functions: \( f(x) = ax^2 + bx + c \)
- Cubic or higher-degree polynomials
We use algebra to expand \( f(x + h) \), subtract \( f(x) \), and simplify the result before taking the limit as \( h \to 0 \).
Key Idea
If a polynomial is differentiable at all points (which it is), this method will always yield a function for the derivative.
Example
Find the derivative of \( f(x) = x^2 \) from first principles.
▶️ Answer/Explanation
$ f'(x) = \lim_{h \to 0} \frac{f(x + h) – f(x)}{h} $
Find \( f(x + h) \):
$ f(x + h) = (x + h)^2 = x^2 + 2xh + h^2 $
Subtract \( f(x) \):
$ f(x + h) – f(x) = (x^2 + 2xh + h^2) – x^2 = 2xh + h^2 $
Divide by \( h \):
$ \frac{f(x + h) – f(x)}{h} = \frac{2xh + h^2}{h} = 2x + h $
Take the limit as \( h \to 0 \):
$ f'(x) = \lim_{h \to 0} (2x + h) = 2x $
Final Answer: \( f'(x) = 2x \)
Higher Derivatives and Notation
Higher Derivatives and Notation
In calculus, once we find the first derivative of a function, we can continue to differentiate to find higher-order derivatives.
First Derivative
The first derivative of a function \( f(x) \) is denoted by:
- \( f'(x) \)
- \( \dfrac{dy}{dx} \) (if \( y = f(x) \))
This represents the slope or rate of change of the function.
Second Derivative
The second derivative is the derivative of the first derivative. It measures the rate of change of the rate of change:
- \( f”(x) \)
- \( \dfrac{d^2y}{dx^2} \)
It provides information about the concavity of a graph and is used to find points of inflection.
Higher Derivatives
We can continue differentiating further. The third derivative, fourth, etc., are denoted as:
- \( f^{(3)}(x), f^{(4)}(x), \ldots \)
- \( \dfrac{d^3y}{dx^3}, \dfrac{d^4y}{dx^4}, \ldots \)
These higher derivatives are used in physics (e.g., acceleration is the second derivative of position) and in analyzing function behavior.
Summary of Notation
| Order | Lagrange Notation | Leibniz Notation |
|---|---|---|
| First Derivative | \( f'(x) \) | \( \dfrac{dy}{dx} \) |
| Second Derivative | \( f”(x) \) | \( \dfrac{d^2y}{dx^2} \) |
| Third Derivative | \( f^{(3)}(x) \) | \( \dfrac{d^3y}{dx^3} \) |
| n-th Derivative | \( f^{(n)}(x) \) | \( \dfrac{d^n y}{dx^n} \) |
Find the first, second, third, and n-th derivatives of \( f(x) = x^4 \)
▶️ Answer/Explanation
$ f'(x) = \dfrac{d}{dx}(x^4) = 4x^3 $
Second Derivative:
$ f”(x) = \dfrac{d^2}{dx^2}(x^4) = \dfrac{d}{dx}(4x^3) = 12x^2 $
Third Derivative:
$ f^{(3)}(x) = \dfrac{d^3}{dx^3}(x^4) = \dfrac{d}{dx}(12x^2) = 24x $
Fourth Derivative:
$ f^{(4)}(x) = \dfrac{d}{dx}(24x) = 24 $
Fifth Derivative and beyond:
$ f^{(n)}(x) = 0 \quad \text{for } n \geq 5 $
Conclusion:
The \( n \)-th derivative of a degree 4 polynomial becomes 0 when \( n > 4 \).
So, $ f^{(n)}(x) = \begin{cases} \dfrac{4!}{(4-n)!}x^{4-n}, & \text{for } 0 \le n \le 4 \\ 0, & \text{for } n > 4 \end{cases} $
