Limits and continuity
The Concept of Limits
A limit describes the behavior of a function as its input approaches a particular value. Formally, we write:
$$\lim_{x \to a} f(x) = L$$
This expression is read as "the limit of $f(x)$ as $x$ approaches $a$ equals $L$." It means that as $x$ gets arbitrarily close to $a$ (but not necessarily equal to $a$), the function values $f(x)$ get arbitrarily close to $L$.
The concept of a limit is fundamental to calculus because it allows us to analyze the behavior of functions at points where they may not be defined or where their behavior changes significantly.
Key Properties of Limits
- A limit may exist even if the function is not defined at that point.
- The limit of a sum equals the sum of the limits: $\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x)$
- The limit of a product equals the product of the limits: $\lim_{x \to a}[f(x) \cdot g(x)] = \lim_{x \to a}f(x) \cdot \lim_{x \to a}g(x)$
- The limit of a quotient equals the quotient of the limits (provided the denominator's limit is not zero): $\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}$
one-sided limits
Sometimes a function approaches different values depending on which direction we approach from:
- Left-hand limit: $\lim_{x \to a^-} f(x)$ (approaching $a$ from values less than $a$)
- Right-hand limit: $\lim_{x \to a^+} f(x)$ (approaching $a$ from values greater than $a$)
A limit exists only when both one-sided limits exist and are equal: $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$.