Limit of a Function: Cauchy's Definition and Its Corollaries

From Sequences to Functions

The limit of a sequence is the limit of a "discrete" function defined on natural numbers. Now we turn to continuous functions, defined on intervals of the real line.

What does it mean that "f(x) tends to L as x tends to a"? Intuition suggests: when x is close to a, the value of f(x) is close to L. But how close? The definition via ε and δ gives a precise answer.

Definition of the Limit (ε-δ)

A number L is called the limit of a function f(x) as x→a if, for any ε > 0, there exists δ > 0 such that for all x with 0 < |x - a| < δ it holds that |f(x) - L| < ε.

Notation: lim(x→a) f(x) = L.

Note: x = a does not enter the condition (0 < |x - a|). The limit describes the behavior of the function near the point, but not at the point itself. This allows us to speak about limits even where the function is not defined.

Example: lim(x→2) (x² - 4)/(x - 2) = ?

When x ≠ 2: (x² - 4)/(x - 2) = (x-2)(x+2)/(x-2) = x + 2. Therefore, the limit = 4, although the original fraction is undefined at x = 2.

One-sided Limits

Left limit (x→a−): we consider x < a. Right limit (x→a+): we consider x > a.

The function |x|/x as x→0: left lim = -1, right lim = +1. The two-sided limit does not exist.

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

Infinite Limits and Limits at Infinity

lim(x→a) f(x) = +∞ means: as x approaches a, f(x) increases without bounds.

lim(x→∞) f(x) = L: as x → +∞, f(x) → L. This is a horizontal asymptote.

Examples: lim(x→∞) (2x+3)/(x-1) = 2; lim(x→0) 1/x² = +∞.

Notable Limits

First: lim(x→0) sin(x)/x = 1.

The proof is geometric: for x ∈ (0, π/2) it holds that sin x < x < tan x. Divide by sin x: 1 < x/sin x < 1/cos x. By the squeeze theorem, sin(x)/x → 1.

Second: lim(x→0) (1+x)^(1/x) = e.

This is the same number e encountered in the limit of sequences.

Both limits are ubiquitous: the first—in the theory of oscillations and optics, the second—in financial mathematics and the theory of growth.

Equivalent Infinitesimals

Functions α(x) and β(x) are equivalent as x→a (α ~ β) if lim α/β = 1.

As x→0: sin x ~ x, tan x ~ x, ln(1+x) ~ x, eˣ - 1 ~ x, (1+x)ⁿ - 1 ~ nx.

These equivalences are a powerful simplification tool. Instead of a complex expression, we substitute an equivalent simple one.

Theorem on the Limit of a Monotonic Function

A monotonic function on an interval has limits from the left and right at every point. The points of discontinuity of a monotonic function form at most a countable set (discontinuities of the first kind).

Connection with Sequences (Heine’s Criterion)

lim(x→a) f(x) = L if and only if for any sequence {xₙ} converging to a (xₙ ≠ a), the sequence {f(xₙ)} converges to L.

Heine’s criterion translates problems about the limits of functions into problems about the limits of sequences and vice versa. This is a powerful tool: to prove that a limit does not exist, it is enough to find two sequences that yield different limits.

Methods of Calculating Limits

In practice, limits are calculated using several key techniques. Algebraic manipulations—multiplying by the conjugate, factoring—eliminate indeterminate forms. Substitution with equivalent infinitesimals simplifies expressions: as x → 0 substitute sin x ~ x, ln(1+x) ~ x, eˣ − 1 ~ x.

Example: lim(x→0) (eˢⁱⁿˣ − 1)/x. As x → 0: sin x ~ x, so eˢⁱⁿˣ − 1 ~ sin x ~ x. The limit equals 1.

Continuity means that the limit can be "brought under the function sign": if g is continuous, then lim g(f(x)) = g(lim f(x)). This greatly simplifies calculations: lim(x→0) cos(sin x) = cos(0) = 1.

Change of variable: lim(x→∞) x sin(1/x) = lim(t→0) sin(t)/t = 1, where t = 1/x.

Limits in Economics and Finance

The limit is a tool for analyzing marginal quantities. Marginal cost MC = dC/dQ = lim(ΔQ→0) ΔC/ΔQ — the derivative of the cost function, and this is the limit of increments. Elasticity coefficient — the ratio of relative changes: E = lim(Δx→0) (Δy/y)/(Δx/x). Portfolio yield calculation with continuous reinvesting relies on the limit (1 + r/n)ⁿ → eʳ.

Question for reflection: Does lim(x→0) x · sin(1/x) exist? What can be said about lim(x→0) sin(1/x)?

L'Hôpital's Rule and Calculation of Indeterminate Forms

If lim f(x) = lim g(x) = 0 or both tend to ∞, and lim f'(x)/g'(x) = L, then lim f(x)/g(x) = L — L'Hôpital's rule. Classical applications: lim(x→0) sin x/x = lim cos x/1 = 1; lim(x→∞) (ln x)/x = lim (1/x)/1 = 0; form 0·∞ is reduced to 0/0 or ∞/∞: lim(x→0⁺) x ln x = lim(x→0⁺) (ln x)/(1/x) = lim(x→0⁺) (1/x)/(−1/x²) = lim(x→0⁺)(−x) = 0. Warning: the rule applies only to genuine indeterminate forms; it is an error to apply it to a limit that is already determinate. The form 1^∞: lim(1 + 1/n)ⁿ = e — calculated by logarithming and L'Hôpital's rule.