1.2/1.3 Limits Grand Teton National Park, Wyoming

Limit notation: “The limit of f of x as x approaches c is L.”

The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See your text for details – p. 59.) For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

does not exist because the left and right hand limits do not match! 2 1 1 2 3 4 At x=1: left hand limit right hand limit value of the function

because the left and right hand limits match. 2 1 1 2 3 4 At x=2: left hand limit right hand limit value of the function

because the left and right hand limits match. 2 1 1 2 3 4 At x=3: left hand limit right hand limit value of the function

Limit of a Polynomial: If p is a polynomial function and c is a real number, then In other words, in the case of polynomials, you may find the limit as x approaches c by using direct substitution.

The Replacement Theorem Let c be a real number and let f(x) = g(x) for all x not equal to c. IF the limit of g(x) exists as x approaches c, then the limit of f(x) also exists and In other words, if you can simplify the given function such that direct substitution will work, do so.

Practice Practice Practice NO CALCULATOR Find: 1.) 2.) 3.) 4.)