Find limits using graphs

Learning Objectives Define a limit in the context of a graph. Determine the limit of a function as x approaches a specific value from the left and right sides using a graph. Evaluate a two-sided limit by comparing the left-hand and right-hand limits. Identify from a graph the three conditions under which a limit does not exist (DNE). Distinguish between the value of a function at a point, f(c), and the limit of the function as x approaches c. Determine limits at infinity by analyzing the end behavior of a graph. Ever wondered how your GPS predicts your arrival time even when you're still miles away? 🗺️ It's all about approaching a destination, which is the core idea of a limit! This tutorial will teach you how to find limits by simply looking at a graph. Understan...

TermDefinitionExample LimitThe y-value that a function *approaches* as the x-value gets closer and closer to some number from both sides. The limit is about the journey, not the destination.On the graph of y = x², as x gets closer to 2 (e.g., 1.9, 1.99, 2.1, 2.01), the y-value gets closer to 4. So, the limit as x approaches 2 is 4. Left-Hand LimitThe y-value a function approaches as x gets closer to a number 'c' from the left side (using values less than c).For a graph with a jump at x=3, if the function approaches y=5 for all x-values just under 3, the left-hand limit is 5. Right-Hand LimitThe y-value a function approaches as x gets closer to a number 'c' from the right side (using values greater than c).For that same graph with a jump at x=3, if the function approach...

Existence of a Two-Sided Limit \lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = L \text{ and } \lim_{x \to c^+} f(x) = L The general (two-sided) limit exists and is equal to L if and only if the left-hand limit and the right-hand limit both exist and are both equal to L. Left-Hand Limit Notation \lim_{x \to c^-} f(x) This notation asks for the y-value the function approaches as x gets closer to 'c' from the left side (values smaller than c). Right-Hand Limit Notation \lim_{x \to c^+} f(x) This notation asks for the y-value the function approaches as x gets closer to 'c' from the right side (values larger than c).