Limit of a function: The value that (f(x)) approaches as (x) approaches a specific point (a). Denoted as (\lim_{x \to a} f(x) = L), meaning (f(x)) gets arbitrarily close to (L) when (x) is sufficiently close to (a).
One-sided limits: The limit of (f(x)) as (x) approaches (a) from the left ((x \to a^-)) or from the right ((x \to a^+)). Both must agree for the two-sided limit to exist.
Infinite limits: When (f(x)) increases or decreases without bound as (x) approaches (a), e.g., (\lim_{x \to a} f(x) = \infty) or (-\infty).
Limit laws: Rules that allow the combination and simplification of limits, such as:
Indeterminate forms: Expressions like (0/0) or (\infty/\infty) that require special techniques (e.g., algebraic manipulation, L'Hôpital's rule) to evaluate limits.
1. What is a limit in calculus?
2. What is the formal limit definition of the derivative of a function at a point?
3. What is the primary role of differentiation techniques in calculus?
Limits — definition?
Values a function approaches near a point.
Derivative — what?
Limit of the average rate of change at a point.
Chain rule — purpose?
Differentiate composite functions efficiently.
Limit of a function — definition?
Value function approaches as x approaches a.
One-sided limits — from where?
Left ( extsuperscript{a-}) or right ( extsuperscript{a+}).
Infinite limits — example?
Function grows without bound near a.
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