2.3 Calculating Limits Using the Limit Laws

Properties of Limits where n is a positive integer. y = c |a|a   (a, c) y = x |a|a   (a, a)

Properties of Limits |a|a |x1|x1 |x2|x2 f (a) = L f (x 1 ) f (x 2 )    Read page 99 – 101 for more properties.  We can use the Direction Substitution Property whenever possible:

Finding limits algebraically 1.Use direct substitution if possible. 2.Simplify the expression first (then substitution): a)Factor and cancel common factors b)Expand and collect like terms (if parentheses are present) c)Rationalize the numerator or denominator (if root is present) 3.Use special limits (if trig functions are present) (HW 3.3) 4.Use one-sided limit (if it is a piecewise function) 5.Use Squeeze Theorem (if sine or cosine is present)

Examples Find the following limit algebraically.

Examples Evaluate the following limits, if they exist.

Examples Find the following limit algebraically. Note we can only use special limits when the variable approaches 0

Squeeze Theorem Therefore, Example Find

Examples Find the following limit algebraically.