Evaluating Limits Numerically & Intro into Algebraic TS: Explicitly assessing information and drawing conclusions Objective: To be able to evaluate a limit graphically and analytically.

General Limit A general limit exists on f (x) when x = c, if the left- and right-hand limits are both equal there. Mathematic Notation: In other words: f (x)  L as x  c

Limits Graphically 6 6 6 6

Limits Graphically Undefined 2 2 2

Limits Graphically 5 8 8 8

Limits Graphically – 3 3 7 DNE

Possible ways to evaluate a limit without a graph. Substitution Factoring Conjugate Table

value is the limit of the Finding Limits = 7 = 7 = 7 If a function approaches the same value from both directions, then that value is the limit of the function at that point. x y x y .9 6.71 1.1 7.31 .99 6.9701 1.01 7.0301 .999 6.997 1.001 7.003

Finding Limits = DNE or NL = 3 = –3 If the Left-hand limit and the Right-hand limit are not equal, the general limit does not exist. x y x y –1.1 3.1 –.9 –2.9 –1.01 3.01 –.99 –2.99 –1.001 3.001 –.999 –2.999

Finding Limits = DNE or NL = -∞ = ∞ x y x y If either the Left-hand limit, Right-hand limit, or both do not exist, the general limit will not exist. 2.9 –44.1 3.1 56.1 2.99 –494 3.01 506.01 2.999 –4994 3.001 5006

How should we best use our calculator to help us if we need to make a table?