Limits Review

Limit L a

c) Find 6 Note: f (-2) = 1 is not involved 2

One-Sided Limit One-Sided Limits The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. L a

The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. M a

Examples of One-Sided Limit 1. Given Find Find

More Examples Find the limits:

A Theorem This theorem is used to show a limit does not exist. For the function

Indeterminate Forms Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Notice form Ex. Factor and cancel common factors

More Examples

Evaluate the limits Answer : 16 Answer : DNE Answer :∞ Answer : 1/2

Limits at Infinity Ex. Degree on top larger: infinity Degree on bottom bigger: 0 Degrees equal: Look at lead coefficients Ex.   =2 =0

Infinite Limits The limit from the left is -∞ The limit from the right is ∞ The limit doesn’t exist The limit from the left is ∞ The limit from the right is ∞ The limit = ∞

Plug in a number to the left of 3 Examples Find the limits   Plug in a number to the left of 3