Limits & Continuity

🔢 Limits & Continuity

Limits and continuity form the rigorous foundation of all calculus. They allow us to talk about the behavior of functions “infinitely close” to a point, even if the function isn’t defined there.

---

🟢 1. The Concept of a Limit

Definition

The limit of a function $f(x)$ as $x$ approaches $c$ is the value $L$ that $f(x)$ gets closer and closer to as $x$ gets closer and closer to $c$ (but not necessarily at $c$). $\lim_{x \to c} f(x) = L$

Computing Limits

• Direct Substitution: If $f(c)$ is defined and $f$ is “nice” (polynomial, rational, etc.), then $\lim_{x \to c} f(x) = f(c)$.
• Factoring and Canceling: For indeterminate forms like $\frac{0}{0}$.
  • Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4$.
• Rationalization: Using the conjugate to simplify square roots.
• L’Hôpital’s Rule: (Advanced) Used for $\frac{0}{0}$ or $\frac{\infty}{\infty}$ using derivatives.

---

🟡 2. Continuity

The Three-Part Definition

A function $f(x)$ is continuous at a point $c$ if and only if all three of the following are true:

1. $f(c)$ is defined.
2. $\lim_{x \to c} f(x)$ exists.
3. $\lim_{x \to c} f(x) = f(c)$.

Types of Discontinuity

• Removable: A “hole” in the graph that can be filled by redefining one point.
• Jump: The left and right limits are different (e.g., step functions).
• Infinite: The function goes to $\pm \infty$ at that point (e.g., $\frac{1}{x}$ at $x=0$).

---

🔴 3. The Intermediate Value Theorem (IVT)

If $f$ is continuous on the closed interval $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in $[a, b]$ such that $f(c) = k$.

Note

The IVT is often used to prove that a function has a root (a zero) on an interval. If $f(a) < 0$ and $f(b) > 0$, and $f$ is continuous, there must be an $x$ between $a$ and $b$ where $f(x) = 0$.

Example Code: Finding a Root numerically

A common way to use IVT in programming is the Bisection Method.

def bisection(f, a, b, tol):
    if f(a) * f(b) >= 0:
        return None # IVT might not apply
    
    while (b - a) / 2.0 > tol:
        midpoint = (a + b) / 2.0
        if f(midpoint) == 0:
            return midpoint
        elif f(a) * f(midpoint) < 0:
            b = midpoint
        else:
            a = midpoint
    return (a + b) / 2.0

# Example: f(x) = x^2 - 2 (finding sqrt(2))
root = bisection(lambda x: x**2 - 2, 1, 2, 0.0001)
print(f"Approximated root: {root}")

---

💡 Key Takeaways

• Limits describe behavior near a point.
• Continuity requires the limit to match the function value.
• IVT guarantees “existence” of values in continuous functions.