Mathematics for Physics I - Limits and Continuity

The Concept of Limits

Limits form the foundation of calculus and are essential for understanding rates of change, continuity, and many physical phenomena.

Informal Definition

The limit of a function f(x) as x approaches a value c is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to c (but not equal to c).

\[ \lim_{x \to c} f(x) = L \]

This means that we can make f(x) as close as we want to L by taking x sufficiently close to c.

Formal (ε-δ) Definition

For every ε > 0, there exists a δ > 0 such that:

\[ \text{if } 0 < |x - c| < \delta \text{ then } |f(x) - L| < \varepsilon \]

This precise definition captures the intuitive idea that we can make f(x) arbitrarily close to L by making x sufficiently close to c.

Interactive visualization of the limit concept

Properties of Limits

Limits obey several important properties that make them easier to work with:

Basic Limit Laws

\[ \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) \] \[ \lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x) \] \[ \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \] \[ \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \text{ if } \lim_{x \to c} g(x) \neq 0 \]

Additional Properties

\[ \lim_{x \to c} k = k \text{ (constant function)} \] \[ \lim_{x \to c} x = c \text{ (identity function)} \] \[ \lim_{x \to c} x^n = c^n \text{ for } n \in \mathbb{Z}^+ \] \[ \lim_{x \to c} \sqrt[n]{x} = \sqrt[n]{c} \text{ for } n \in \mathbb{Z}^+ \text{ and } c > 0 \text{ if } n \text{ is even} \]

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) for all x near c (except possibly at c) and:

\[ \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L \]

Then:

\[ \lim_{x \to c} f(x) = L \]

This is particularly useful for finding limits of oscillating functions.

One-Sided Limits

The left-hand limit is denoted:

\[ \lim_{x \to c^-} f(x) = L \]

The right-hand limit is denoted:

\[ \lim_{x \to c^+} f(x) = M \]

The limit exists if and only if both one-sided limits exist and are equal:

\[ \lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L \]

Techniques for Finding Limits

Direct Substitution

If f is continuous at c, then:

\[ \lim_{x \to c} f(x) = f(c) \]

Example:

\[ \lim_{x \to 2} (x^2 + 3x) = 2^2 + 3(2) = 4 + 6 = 10 \]

Factoring

Useful for limits of the form \(\frac{0}{0}\):

\[ \lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3) = 6 \]

Rationalization

Useful for limits involving radicals:

\[ \lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} = \lim_{x \to 4} \frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{(x - 4)(\sqrt{x} + 2)} = \lim_{x \to 4} \frac{x - 4}{(x - 4)(\sqrt{x} + 2)} = \lim_{x \to 4} \frac{1}{\sqrt{x} + 2} = \frac{1}{4} \]

L'Hôpital's Rule

For limits of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\):

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]

Provided the limit on the right exists or is infinite.

Important Limits

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \] \[ \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \] \[ \lim_{x \to 0} (1 + x)^{1/x} = e \] \[ \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \]

These limits are particularly important in physics applications, especially in the analysis of oscillatory systems and exponential growth or decay.

Continuity

Continuity is a fundamental concept in calculus that describes functions without "breaks" or "jumps".

Definition of Continuity

A function f is continuous at a point c if:

f(c) is defined
\(\lim_{x \to c} f(x)\) exists
\(\lim_{x \to c} f(x) = f(c)\)

In other words, a function is continuous at c if the limit of the function as x approaches c equals the function value at c.

Types of Discontinuities

Removable Discontinuity: The limit exists but is not equal to the function value (or the function is not defined at that point)
Jump Discontinuity: The left and right limits exist but are not equal
Infinite Discontinuity: The limit approaches infinity
Oscillatory Discontinuity: The function oscillates infinitely as x approaches c

Visualization of different types of discontinuities

Properties of Continuous Functions

Intermediate Value Theorem: If f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists at least one value c in [a, b] such that f(c) = k.
Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains both a maximum and minimum value on [a, b].
Composition of Continuous Functions: If g is continuous at c and f is continuous at g(c), then the composition (f ∘ g) is continuous at c.

Applications in Physics

Instantaneous Velocity

The instantaneous velocity of an object is defined as the limit of the average velocity as the time interval approaches zero:

\[ v(t) = \lim_{\Delta t \to 0} \frac{x(t + \Delta t) - x(t)}{\Delta t} \]

This is the fundamental concept behind derivatives and is essential for analyzing motion.

Electric Field

The electric field at a point is defined as the limit of the force per unit charge as the test charge approaches zero:

\[ \vec{E} = \lim_{q \to 0} \frac{\vec{F}}{q} \]

This ensures that the test charge doesn't significantly disturb the field being measured.

Thermodynamic Limits

Many thermodynamic quantities are defined as limits. For example, the heat capacity at constant volume:

\[ C_V = \lim_{\Delta T \to 0} \frac{\Delta Q}{\Delta T} \]

This allows us to define instantaneous rates of change in thermodynamic systems.

Wave Phenomena

The continuity of wave functions is essential in quantum mechanics and classical wave theory. Discontinuities in wave functions often represent physical boundaries or potential barriers.

For example, the continuity of the wave function and its derivative at boundaries leads to reflection and transmission coefficients in quantum mechanics.

Example: Analyzing a Physical System

Consider a mass on a spring with position function:

\[ x(t) = A\cos(\omega t + \phi) \]

The velocity function is:

\[ v(t) = \lim_{\Delta t \to 0} \frac{x(t + \Delta t) - x(t)}{\Delta t} = -A\omega\sin(\omega t + \phi) \]

The continuity of the position and velocity functions ensures that the motion is smooth and physically realistic. Any discontinuity would represent an unphysical "jump" in position or an infinite acceleration.

Key Insight:

Limits and continuity provide the mathematical foundation for describing smooth changes in physical systems. They allow us to define instantaneous rates of change, which are essential for formulating the fundamental laws of physics. The concept of continuity is deeply connected to the idea that physical processes generally occur smoothly, without instantaneous jumps in physical quantities.

Practice Problems

Test your understanding of limits and continuity with these practice problems:

Evaluate \(\lim_{x \to 2} \frac{x^3 - 8}{x - 2}\).
Find \(\lim_{x \to 0} \frac{\sin(3x)}{x}\).
Determine where the function \(f(x) = \frac{x^2 - 4}{x - 2}\) is continuous. Identify and classify any discontinuities.
Use the Squeeze Theorem to evaluate \(\lim_{x \to 0} x^2\sin\left(\frac{1}{x}\right)\).
A particle moves along a straight line with position function \(s(t) = t^3 - 6t^2 + 9t + 1\). Find the instantaneous velocity at t = 2 using the limit definition.