5  Limits of Functions

5.1 Summary Limits of Functions

Limits reveal how functions behave as inputs get closer to a given value. They are essential in calculus, laying the groundwork for derivatives and continuity. With limits, we can explore instantaneous change, refine approximations, and resolve problems where direct evaluation fails. An overview of the main ideas and applications appears in Table 5.1.

Table 5.1: Key Concepts in Limits and Continuity

KeyConcept	Description	ExampleApplication
Definition	\(\lim_{x \to a} f(x) = L\); the value \(f(x)\) approaches as \(x \to a\)	Approximating function values: \(\lim_{x \to 0} \tfrac{\sin x}{x} = 1\)
One-sided Limits	Limits approaching from left (\(x \to a^-\)) or right (\(x \to a^+\))	Instantaneous velocity from left/right time intervals
Continuity	\(f\) is continuous at \(x=a\) if \(\lim_{x \to a} f(x) = f(a)\)	Ensuring smooth motion or consistent output in physical systems