1 3 Evaluating Limits Analytically Evaluate a limit

1. 3 Evaluating Limits Analytically • Evaluate a limit using properties of limits. • Develop and use a strategy for finding limits. • Evaluate a limit using dividing out and rationalizing techniques. • Evaluate a limit using the Squeeze Theorem.

lim 3 x = How to read the notation. The limit of 3 x as x approaches 4 is______ Some basic limits 1. Lim b = b 2. Lim x = c 3. Lim xⁿ = cⁿ

Algebraically 1. Direct Substitution- always try first because it is so easy. Plug in the “c” value into x and simplify. This works for “well-behaved” functions that are continuous at c. a) Polynomial If p is a polynomial function and c is a real number, the lim p(x) = p(c). b) Rational If r is a rational function given by r(x) = p(x)/q(x) and c is a real number such that q(c) ≠ 0, then lim r(x) = r(c) = p(c)/q(c).

c) Radical lim x = c , This limit is valid for all c if n is odd, and is valid for c > 0 if n is even. d) Composite Function Find the limit of the inside function first then evaluate the outside function. e) Trigonometric Functions Let c be a real number in the domain of the given trigonometric function. lim sin x = sin c lim cos x = cos c lim tan x = tan c lim cot x = cot c lim sec x = sec c lim csc x = csc c

Examples

If you try substitution and get an undefined value, #/0 or 0/0. Then you will have to use another method to rewrite the function. 2. Factoring

3. Rationalizing the numerator

The Squeeze Theorem It squeezes a function, that you can not find the limit of using another method, between two functions that you can find the limit of at c. This theorem was used to find the following to special trigonometric limits. 1) lim 2) lim