Calculating Limits Using The Limit Laws

2 Basic Limit Laws where n is a positive integer. y = c |a|a   (a, c) y = x |a|a   (a, a) where n is a positive integer.

3 Limit Laws Generalized Suppose that c is a constant and the following limits exist

4 Limit Laws Generalized where n is a positive integer.

5 Examples Evaluate the following limits. Justify each step using the laws of limits.

6 Direct Substitution Property If f is a polynomial or a rational function and a is in the domain of f, then

7 Examples 1.If f is a rational function or complex: a.Simplify the function; eliminate common factors. b.Find a common denominator. c.Perform long division. 2.If a root function exists, rationalize the numerator or denominator. 3.If an absolute values function exists, use one-sided limits and the definition. You may encounter limit problems that seem to be impossible to compute or they appear to not exist. Here are some tricks to help you evaluate these limits.

8 Direct Substitution Property Evaluate the following limits, if they exist.

9 Theorem If f(x)  g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then

10 The Squeeze (Sandwich) Theorem If f(x)  g(x)  h(x) when x is near a (except possibly at a) and then

11 Example Prove that is true. Strategy To begin, bind a part of the function (usually the trigonometric part if present) between two real numbers. Then create the original function in the middle.

12 You Try It Evaluate the following limits, if they exist, in groups of no more than three members.