AP Calculus Chapter 1, Section 3 Evaluating Limits Analytically 2013 - 2014

Some Basic Limits In some cases, the limit can be evaluated by direct substitution. lim 𝑥→𝑐 𝑏 =𝑏 lim 𝑥→𝑐 𝑥 =𝑐 lim 𝑥→𝑐 𝑥 𝑛 = 𝑐 𝑛

Evaluate the following limits

Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits. lim 𝑥→𝑐 𝑓 𝑥 =𝐿 and lim 𝑥→𝑐 𝑔 𝑥 =𝐾 Scalar multiple: lim 𝑥→𝑐 𝑏𝑓 𝑥 =𝑏𝐿 Sum or difference: lim 𝑥→𝑐 𝑓 𝑥 ±𝑔 𝑥 =𝐿±𝐾 Product: lim 𝑥→𝑐 𝑓 𝑥 𝑔 𝑥 =𝐿𝐾 Quotient: lim 𝑥→𝑐 𝑓(𝑥) 𝑔(𝑥) = 𝐿 𝐾 ′ provided 𝐾≠0 Power: lim 𝑥→𝑐 [𝑓 𝑥 ] 𝑛 = 𝐿 𝑛

lim 𝑥→2 ( 4𝑥 2 +3)

Limits of Polynomials & Rational Functions If p is a polynomial function and c is a real number, then lim 𝑥→𝑐 𝑝 𝑥 =𝑝(𝑐) If r is a rational function given by r 𝑥 =𝑝(𝑥)/𝑞(𝑥) and c is a real number such 𝑞(𝑐)≠0, then lim 𝑥→𝑐 𝑟 𝑥 =𝑟 𝑐 = 𝑝(𝑐) 𝑞(𝑐)

lim 𝑥→1 𝑥 2 +𝑥+2 𝑥+1

The Limit of a Function Involving a Radical Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for 𝑐>0 if n is even. lim 𝑥→𝑐 𝑛 𝑥 = 𝑛 𝑐

The Limit of a Composite Function If f and g are functions such that lim 𝑥→𝑐 𝑔 𝑥 =𝐿 and lim 𝑥→𝑐 𝑓 𝑥 =𝑓(𝐿) , then lim 𝑥→𝑐 𝑓 𝑔 𝑥 =𝑓 lim 𝑥→𝑐 𝑔 𝑥 =𝑓(𝐿)

Because lim 𝑥→0 𝑥 2 +4 = 0 2 +4=4 and lim 𝑥→4 𝑥 =2 it follows that lim 𝑥→0 𝑥 2 +4 = 4 =2

Limits of Trigonometric Functions lim 𝑥→𝑐 sin 𝑥 = sin 𝑐 lim 𝑥→𝑐 cos 𝑥 = cos 𝑐 lim 𝑥→𝑐 tan 𝑥 = tan 𝑐 lim 𝑥→𝑐 cot 𝑥 = cot 𝑐 lim 𝑥→𝑐 csc 𝑥 = csc 𝑐 lim 𝑥→𝑐 sec 𝑥 = sec 𝑐

lim 𝑥→0 tan 𝑥 lim 𝑥→𝜋 (𝑥 cos 𝑥 ) lim 𝑥→0 sin 2 x

Dividing Out Technique lim 𝑥→−3 𝑥 2 +𝑥−6 𝑥+3

Rationalizing Technique lim 𝑥→0 𝑥+1 −1 𝑥 Check your answer by using a table

Then lim 𝑥→𝑐 𝑓(𝑥) exists and is equal to L. The Squeeze Theorem Basically says if you have two different function that have the same limit as 𝑥→𝑐, and you have a 3rd function that falls between the first two functions, the 3rd function will also have the same limit as 𝑥→𝑐. ℎ(𝑥)≤𝑓 𝑥 ≤𝑔 𝑥 for all x in an open interval containing c, except possibly c itself, and if lim 𝑥→𝑐 ℎ 𝑥 =𝐿= lim 𝑥→𝑐 𝑔(𝑥) Then lim 𝑥→𝑐 𝑓(𝑥) exists and is equal to L.

Special Trigonometric Functions lim 𝑥→0 sin 𝑥 𝑥 =1 lim 𝑥→0 1− cos 𝑥 𝑥 =0

Find the limit: lim 𝑥→0 tan 𝑥 𝑥

Find the limit: lim 𝑥→0 sin 4𝑥 𝑥

Homework Pg. 67 – 69: #1 – 77 every other odd, 83, 87, 113