Limits and Limit Laws in Calculus: Study Notes

Limits and Limit Laws

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding limits is essential for studying continuity, derivatives, and integrals.

• Limit of a Function: The value that a function approaches as the input approaches a specific point.

• Notation: $\lim_{x \to a} f(x)$ denotes the limit of $f(x)$ as $x$ approaches $a$.

• Key Properties: Limits can often be evaluated using algebraic manipulation and limit laws.

Limit Laws

Limit laws allow us to evaluate limits of combinations of functions using the limits of the individual functions. These laws are valid when the individual limits exist.

• Sum Law: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$

• Product Law: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

• Quotient Law: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$

• Root Law: $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$, provided the limit inside the root exists and is valid for real roots.

Evaluating Limits: Example Problems

Let $f(x)$ and $g(x)$ be functions such that $\lim_{x \to 3} f(x) = 5$ and $\lim_{x \to 3} g(x) = -2$. We use limit laws to evaluate the following:

Example 1: Sum of Functions

Problem: $\lim_{x \to 3} [f(x) + 7g(x)]$

• Apply the sum and constant multiple laws:

$\lim_{x \to 3} [f(x) + 7g(x)] = \lim_{x \to 3} f(x) + 7 \lim_{x \to 3} g(x)$

$= 5 + 7(-2) = 5 - 14 = -9$

• Final Answer: $-9$

Example 2: Quotient Involving Roots

Problem: $\lim_{x \to 3} \frac{f(x)}{\sqrt{f(x)} + g(x)^2}$

• Evaluate each part using the given limits:

$\lim_{x \to 3} f(x) = 5$

$\lim_{x \to 3} g(x) = -2 \implies \lim_{x \to 3} g(x)^2 = (-2)^2 = 4$

$\lim_{x \to 3} \sqrt{f(x)} = \sqrt{5}$

$\lim_{x \to 3} [\sqrt{f(x)} + g(x)^2] = \sqrt{5} + 4$

$\lim_{x \to 3} \frac{f(x)}{\sqrt{f(x)} + g(x)^2} = \frac{5}{\sqrt{5} + 4}$

• Final Answer: $\frac{5}{\sqrt{5} + 4}$

Summary Table: Limit Laws Used

Law	Formula	Application
Sum Law	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$	Used to split limits of sums
Constant Multiple Law	$\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$	Used for $7g(x)$ term
Root Law	$\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$	Used for $\sqrt{f(x)}$
Quotient Law	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$	Used for the entire fraction

Additional info:

• These problems are typical of early calculus courses, focusing on the application of limit laws to evaluate limits without needing the explicit form of the functions.

• Mastery of these laws is essential for progressing to more advanced topics such as continuity and differentiation.