Definition of the Limit and Limit Theorems

Tags: #◇Computer Science #◇CSMATH1 #◇Problem Solving

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◇Overview

Precise Definition of a Limit

The precise definition of a limit (often called the epsilon-delta definition) states that $\lim_{x \to c} f(x) = L$ if, for every number $\epsilon > 0$, there exists a corresponding number $\delta > 0$ such that:
$0 < |x - c| < \delta \implies |f(x) - L| < \epsilon$
This means we can make the value of $f(x)$ as close to $L$ as we want (within $\epsilon$) by choosing $x$ sufficiently close to $c$ (within $\delta$, but not equal to $c$).

Limit Theorems

To evaluate limits without relying on tables or graphs, we use established Limit Theorems:

1. Limit of a Constant: $\lim_{x \to a} k = k$
2. Substitution Law: $\lim_{x \to a} x = a$
3. Addition Theorem: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
4. Constant Multiple Theorem: $\lim_{x \to a} [k \cdot g(x)] = k \cdot \lim_{x \to a} g(x)$
5. Multiplication Theorem: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
6. Division Theorem: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$
7. Power Theorem: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$
8. Radical Root Theorem: $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$, provided the $n$-th root is a real number (if $n$ is even, the limit must be positive).

Limits of Functions

Polynomial Function: $\lim_{x \to a} p(x) = p(a)$
Rational Function: $\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}$, provided $q(a) \neq 0$

◇Important Formulas

Epsilon-Delta Definition: $0 < |x - c| < \delta \implies |f(x) - L| < \epsilon$
Polynomial Limit: $\lim_{x \to a} f(x) = f(a)$ for any polynomial $f$.

◇Worked Examples

Example 1: Using Limit Theorems
Find $\lim_{x \to 2} (x^2 + 2x - 3)$.
Solution:
$\lim_{x \to 2} (x^2 + 2x - 3) = \lim_{x \to 2} x^2 + \lim_{x \to 2} 2x - \lim_{x \to 2} 3 = 2^2 + 2(2) - 3 = 4 + 4 - 3 = 5$

Example 2: Limit of a Rational Function (Direct Substitution)
Find $\lim_{x \to 4} \frac{x^2 - 16}{2x - 8}$.
Solution:
Direct substitution yields $\frac{0}{0}$. We must factor and simplify:
$\lim_{x \to 4} \frac{(x + 4)(x - 4)}{2(x - 4)} = \lim_{x \to 4} \frac{x + 4}{2} = \frac{4 + 4}{2} = \frac{8}{2} = 4$

Example 3: Limit of a Radical Function
Find $\lim_{x \to 2} \sqrt{\frac{x^3 + 2x + 3}{x^2 + 5}}$.
Solution:
$\lim_{x \to 2} \frac{x^3 + 2x + 3}{x^2 + 5} = \frac{2^3 + 2(2) + 3}{2^2 + 5} = \frac{8 + 4 + 3}{4 + 5} = \frac{15}{9} = \frac{5}{3}$
So, the limit is $\sqrt{\frac{5}{3}}$.

Example 4: Limit of a Piecewise Function
Given $f(x) = \begin{cases} x^5 - 12 & \text{for } x < 2 \\ (x+3)^3 - 8 & \text{for } x \ge 2 \end{cases}$. Find $\lim_{x \to 2} f(x)$.
Solution:
Left limit: $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^5 - 12) = 2^5 - 12 = 32 - 12 = 20$
Right limit: $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} ((x+3)^3 - 8) = (2+3)^3 - 8 = 125 - 8 = 117$
Since $20 \neq 117$, the limit does not exist (DNE).
(Note: OCR indicated answer 19 for right limit, $(2+3)^3 - 8 = 125 - 8 = 117$. If it was $\frac{1}{(x+3)^3-8}$, it's different. In OCR: $(x + 3)^3 - 8$, wait, OCR actually says [1] and = 19 maybe? But 20 != 19. Just state DNE.)

◇References & Navigation

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• Related Notes: ◇Introduction to Limits, ◇Continuity