Lecture 11: As X Approaches...

Functions and limits — the mathematics of nearness

The Chemical Hook: When two argon atoms approach each other, what happens to the potential energy? At large distances, they barely interact. At intermediate distances, they attract. At very short distances, they repel violently. The whole story of chemistry is written in this curve of "as r approaches..."

1. Functions: The Language of Dependence

1.1 What Is a Function?

A function is a rule that assigns to each input exactly one output. We write:

$$f: X \to Y$$

$$x \mapsto f(x)$$

"For each element x in the domain X, there is exactly one element f(x) in the codomain Y."

Definition (Function): A function f from set X to set Y is a rule that assigns to each element x ∈ X a unique element f(x) ∈ Y. X is the domain, Y is the codomain, and the set of all outputs {f(x) : x ∈ X} is the range.

1.2 Functions in Chemistry

Input (Domain) Output (Range) Function
Distance r between atoms Potential energy V V(r) — Lennard-Jones potential
Substrate concentration [S] Reaction rate v v([S]) — Michaelis-Menten
Temperature T Fraction in excited state Boltzmann distribution
Time t Concentration [A] [A](t) — kinetics

Interactive: Function Grapher

3x
Domain
ℝ
Range
[0, ∞)
Zeros
x = 1

1.3 Key Properties of Functions

2. Limits: The Mathematics of Approach

2.1 The Intuition

The limit captures what happens as we get arbitrarily close to a point, without necessarily reaching it.

Question: What is the value of sin(x)/x when x = 0?

Direct substitution gives 0/0 — undefined! But watch what happens as x approaches 0...

Interactive: Limit Explorer

1.00
f(x) = sin(x)/x
0.841
Limit as x→0
1.000
Distance from limit
0.159

2.2 Formal Definition

Definition (Limit): We say limx→a f(x) = L if for every ε > 0, there exists δ > 0 such that:

$$0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$$

"For any desired closeness ε to L, we can find a neighborhood δ around a where f(x) stays within ε of L."

2.3 One-Sided Limits

Sometimes behavior differs from the left and right:

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

3. Limit Laws

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

Operation Result
Sum limx→a [f(x) + g(x)] = L + M
Difference limx→a [f(x) - g(x)] = L - M
Product limx→a [f(x) · g(x)] = L · M
Quotient limx→a [f(x) / g(x)] = L / M (if M ≠ 0)
Constant multiple limx→a [c · f(x)] = c · L
Power limx→a [f(x)]ⁿ = Lⁿ

4. Continuity: When Limits Behave

4.1 Definition

Definition (Continuity): A function f is continuous at a if:
  1. f(a) is defined
  2. limx→a f(x) exists
  3. limx→a f(x) = f(a)

In words: the function value equals the limiting value. No jumps, no gaps, no surprises.

Interactive: Types of Discontinuity

Type
Continuous
Left limit
—
Right limit
—

4.2 Physical Significance

Most physical quantities are continuous. Nature doesn't jump:

Discontinuities signal special physics: phase transitions, boundaries, or singularities.

5. Chemical Applications

5.1 The Lennard-Jones Potential

The potential energy between two neutral atoms:

$$V(r) = 4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right]$$

Interactive: Lennard-Jones Potential

1.0
1.0
lim V(r) as r→0
+∞
lim V(r) as r→∞
0
Minimum at
r = 1.12σ

Limit analysis:

5.2 Michaelis-Menten Kinetics

Enzyme-catalyzed reaction rate as a function of substrate concentration:

$$v = \frac{V_{max}[S]}{K_M + [S]}$$

Interactive: Michaelis-Menten Kinetics

1.0
0.5
lim v as [S]→0
0
lim v as [S]→∞
Vmax
v at [S] = KM
Vmax/2

5.3 Boltzmann Distribution

Fraction of molecules in excited state as a function of temperature:

$$\frac{n_i}{N} = \frac{e^{-E_i/k_BT}}{Z}$$

Interactive: Boltzmann Distribution

300 K
2 kT
lim P(excited) as T→0
0
lim P(excited) as T→∞
1/2
Current P(excited)
—

6. Exercises

Exercise 1: Evaluating Limits

Evaluate the following limits:

  1. limx→2 (x² - 4)/(x - 2)
  2. limx→0 (ex - 1)/x
  3. limx→∞ (3x² + 2x)/(x² + 1)
  1. (x² - 4)/(x - 2) = (x+2)(x-2)/(x-2) = x + 2 → 4
  2. This is the definition of the derivative of ex at x=0 → 1
  3. Divide top and bottom by x²: (3 + 2/x)/(1 + 1/x²) → 3

Exercise 2: Lennard-Jones Analysis

For the Lennard-Jones potential V(r) = 4ε[(σ/r)¹² - (σ/r)⁶]:

  1. Find the equilibrium distance req where dV/dr = 0
  2. What is V(req)?
  3. At what distance r₀ does V(r₀) = 0?
  1. Set dV/dr = 0: 4ε[-12σ¹²/r¹³ + 6σ⁶/r⁷] = 0 → req = 21/6σ ≈ 1.122σ
  2. V(req) = 4ε[1/4 - 1/2] = -ε
  3. V(r₀) = 0 when (σ/r)¹² = (σ/r)⁶ → r₀ = σ

Exercise 3: Michaelis-Menten

An enzyme has Vmax = 100 μM/s and KM = 5 μM.

  1. What is v when [S] = 5 μM?
  2. What [S] gives v = 90 μM/s?
  3. Explain why v approaches Vmax but never exceeds it.
  1. v = 100·5/(5+5) = 50 μM/s (half of Vmax)
  2. 90 = 100·[S]/(5+[S]) → 90(5+[S]) = 100[S] → [S] = 45 μM
  3. As [S]→∞, v = Vmax·[S]/([S](KM/[S]+1)) → Vmax. The enzyme is the bottleneck; even infinite substrate can't make it go faster than Vmax.

Exercise 4: Continuity

Determine if the following functions are continuous at the given point:

  1. f(x) = |x|/x at x = 0
  2. f(x) = (x² - 9)/(x - 3) at x = 3
  3. f(x) = e-1/x² for x ≠ 0, f(0) = 0 at x = 0
  1. Not continuous: limx→0⁺ = 1, limx→0⁻ = -1 (jump discontinuity)
  2. Not continuous (removable): f(3) undefined, but limx→3 = 6
  3. Continuous: limx→0 e-1/x² = 0 = f(0) (smooth function)

Coming Up: Lecture 12

Derivatives — When limits meet rates of change

The CHANGE primitive: how functions transform

References: McQuarrie, Mathematics for Physical Chemistry; Atkins, Physical Chemistry