The Theory of Limits & Rigorous ε-δ Analysis

1. Intuitive Limit Concept

In calculus, the limit is the foundational concept that enables the transition from static algebra to dynamical rates of change. Intuitively, we say that the limit of $f(x)$ as $x$ approaches $c$ is $L$:

if the values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$ (with $x \ne c$). Crucially, the value of $f(c)$ itself is completely irrelevant to the limit; $f(c)$ may equal $L$, may equal some other value, or may be completely undefined.

2. One-Sided Limits

Often, a function behaves differently depending on whether $x$ approaches $c$ from numbers strictly less than $c$ (from the left) or strictly greater than $c$ (from the right):

Theorem (Two-Sided Limit Existence Criterion): The two-sided limit $\lim_{x \to c} f(x)$ exists and equals $L$ if and only if both one-sided limits exist and are strictly equal:

If $\lim_{x \to c^-} f(x) \ne \lim_{x \to c^+} f(x)$, the two-sided limit does not exist (DNE).

1. The Formal Cauchy-Weierstrass Definition

To eliminate ambiguity in phrases like "arbitrarily close" and "sufficiently close", Augustin-Louis Cauchy and Karl Weierstrass formulated the rigorous $\varepsilon$-$\delta$ definition:

2. Geometric Interpretation of the $\varepsilon$-$\delta$ Challenge

The definition is a mathematical game between two players:

1. A challenger proposes a tolerance $\varepsilon > 0$, demanding that $f(x)$ lie in the horizontal target band $(L - \varepsilon, L + \varepsilon)$.
2. You must respond by finding a neighborhood radius $\delta > 0$ such that for every $x$ within the punctured vertical interval $(c - \delta, c + \delta) \setminus \{c\}$, the graph $y = f(x)$ stays strictly inside the horizontal $\varepsilon$-tube.

Notice that $\delta$ depends on both $\varepsilon$ and $c$: $\delta = \delta(\varepsilon, c)$.

3. The Canonical $\varepsilon$-$\delta$ Proof Template

A rigorous $\varepsilon$-$\delta$ proof always proceeds in two distinct phases:

• Scratchwork Analysis: Start with the target inequality $|f(x) - L| < \varepsilon$. Factor out $|x - c|$ to obtain $|x - c| \cdot |g(x)| < \varepsilon$. Choose a preliminary bound on $|x - c|$ (typically $\delta_1 \le 1$) to bound the auxiliary factor $|g(x)| \le M$. Then deduce $\delta = \min\{1, \varepsilon / M\}$.
• Formal Deductive Proof: State "Let $\varepsilon > 0$ be given. Choose $\delta = \min\{1, \varepsilon / M\}$." Then, assuming $0 < |x - c| < \delta$, deduce sequentially that $|f(x) - L| < \varepsilon$ by direct algebraic inequalities.

1. Limit Arithmetic Theorems

Let $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$. Then:

2. The Squeeze (Sandwich) Theorem

3. Rigorous Geometric Proof of the Fundamental Limit $\lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1$

Consider a unit circle of radius $r = 1$. For an acute central angle $\theta \in (0, \pi/2)$ measured in radians:

1. Area of inner triangle with vertices $(0,0), (1,0), (\cos\theta, \sin\theta)$:
   $$A_{\text{inner}} = \frac{1}{2} (1) (\sin\theta) = \frac{1}{2} \sin\theta$$
2. Area of circular sector of angle $\theta$:
   $$A_{\text{sector}} = \frac{1}{2} r^2 \theta = \frac{1}{2} \theta$$
3. Area of outer tangent triangle with vertices $(0,0), (1,0), (1, \tan\theta)$:
   $$A_{\text{outer}} = \frac{1}{2} (1) (\tan\theta) = \frac{1}{2} \frac{\sin\theta}{\cos\theta}$$

By geometric containment, $A_{\text{inner}} < A_{\text{sector}} < A_{\text{outer}}$:

Multiplying by $2 / \sin\theta$ (since $\sin\theta > 0$ on $(0, \pi/2)$):

Taking the limit as $\theta \to 0^+$: since $\lim_{\theta \to 0^+} \cos\theta = 1$, the Squeeze Theorem forces $\lim_{\theta \to 0^+} \frac{\sin\theta}{\theta} = 1$. By even symmetry $\frac{\sin(-\theta)}{-\theta} = \frac{\sin\theta}{\theta}$, the two-sided limit is established: $\lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1 \quad \blacksquare$

1. Limits at Infinity & Horizontal Asymptotes

If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, the horizontal line $y = L$ is a horizontal asymptote of the graph $y = f(x)$.

2. Fundamental Rational Limit Theorem

For any rational power $r > 0$:

3. Infinite Limits & Vertical Asymptotes

If $f(x) \to \pm\infty$ as $x \to c^+$ or $x \to c^-$, the line $x = c$ is a vertical asymptote.