Limits and Graphs Overview e.2

Overview

This lecture reviews the concept of limits, one-sided limits, and how to read limits from graphs, highlighting cases where limits do or do not exist.

The limit of (f(x)) as (x) approaches (a) is (L) if (f(x)) can be made arbitrarily close to (L) by taking (x) close to but not equal to (a).
Standard notation: (\lim_{x \to a} f(x) = L).
Limits focus on what happens as (x) approaches (a), not necessarily at (a) itself.

Right-hand limit: (\lim_{x \to a^+} f(x) = L), considers (x > a).
Left-hand limit: (\lim_{x \to a^-} f(x) = L), considers (x < a).
The two-sided limit exists if and only if both one-sided limits exist and are equal.

For (f(x) = x/|x|), as (x \to 0^-) the limit is (-1); as (x \to 0^+) it is (1); the two-sided limit does not exist.
For (f(x) = \sqrt{9-x^2}), as (x \to 3^-) the limit is (0); as (x \to 3^+), the function is not defined, so the limit does not exist.
For (f(x) = \sin(\pi/x)), as (x \to 0) the function oscillates infinitely between (-1) and (1); the limit does not exist.

To find (\lim_{x \to a} f(x)) from a graph, check the value (f(x)) approaches from both sides as (x) nears (a).
If both sides approach the same value, that is the limit; if they differ, the limit does not exist.
Example: If the graph approaches (y=3) from both sides as (x \to 2), then (\lim_{x \to 2} f(x) = 3).
The value of the function at (x = a) ((f(a))) may be different from the limit as (x \to a).
Vertical asymptotes can result in limits approaching infinity, which means the limit does not exist.

Limit — The value a function approaches as the input approaches a specific point.
One-Sided Limit — The value a function approaches from only the left or right direction.
Two-Sided Limit — The value a function approaches from both directions, exists only if both one-sided limits are equal.
Undefined — The function has no value at that input.
Vertical Asymptote — A line where the function increases or decreases without bound as the input approaches a certain value.

Practice reading limits from graphs.
Prepare for discussion on limits approaching infinity and further techniques for evaluating limits without graphs.