2.1 Introduction to Limits Rita Korsunsky

Limit of a function • Notation: ° • L Intuitive Meaning: x a X

Example 1: f(x) = x+2 x 0.9 2.9 f(x) = x + 2 3 2 1 0.99 2.99 0.999 2.999 1.1 3.1 1.01 3.01 1.001 3.001 = 3

Example 2 3 2 1 Even though g(1)  3, the limit is still 3 y = x +2 From example1= 3 Even though g(1)  3, the limit is still 3

One-sided Limits   Notation: y = f(x)  Intuitive Meaning: We can make f(x) as close to L as desired by choosing x sufficiently close to a, and x < a.  f(x) L x a Notation: y = f(x)  Intuitive Meaning: We can make f(x) as close to L as desired by choosing x sufficiently close to a, and x >a.  f(x) L a x

Theorem if and only if

Example 3 As the graph shows, f(x) does not approach a specific number L as x approaches 0 from the right and from the left, limit does not exist.

Example 4 y D.N.E x

Example 5 y -1 1 x D.N.E (left-hand and right-hand limits are not equal)

Example 6 1 0 1 DNE 1 1 1 2 2 2 1 Given y = f(x), find the limits from x = 0 to 4. 1 At x = 0: At x = 1: At x = 2: At x = 3: At x = 4: 0 1 DNE Even though f(1) = 1 1 1 1 Even though f(2) = 2 2 2 2 1

Example 7

Example 8