1. AP Calculus Chapter 1, Section 3
Evaluating Limits Analytically

2. 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

3. Evaluate the following limits

4. 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

5. lim 𝑥→2 ( 4𝑥 2 +3)

6. 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)

7. lim 𝑥→1 𝑥 2 +𝑥+2 𝑥+1

8. 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

9. 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)

10. 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

11. 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

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

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

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

15. 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.

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

17. Find the limit: lim 𝑥→0 tan 𝑥 𝑥

18. Find the limit: lim 𝑥→0 sin 4𝑥 𝑥

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