Some Basic Limits

1
Some Basic Limits

• Limx-gtc b b
• Limx-gtc x c
• Limx-gtcxn cn

2
Properties of Limits

• Let b and c be real numbers, n a positive
  integer, and F and G functions so that
• Limx-gtcF(X) L and
• Limx-gtcG(x) K.

3
Then

• Limx-gtcbF(x) bL
• Limx-gtcF(x) ? G(x) L ? K
• Limx-gtcF(x)G(x) LK
• Limx-gtc F(x)/G(X) L/K, if K ? 0
• Limx-gtc F(x)n Ln

4
Two more Properties

• If p(x) is a polynomial, then limx-gtcp(x) p(c).
  
• If r(x) p(x)/q(x) is a rational function and
  q(c) ? 0, then limx-gtcp(x)/q(x) p(c)/q(c).

5
Composite Functions

• If F and G are functions so that limx-gtcG(x) L
  and limx-gtc F(x) F(L) then
• limx-gtc F(G(x)) F(L)

6
Trigonometric Functions

• Limx-gtc sin(x) sin(c)
• Limx-gtc cos(x) cos(c)
• Limx-gtc tan(x) tan(c)
• Limx-gtc cot(x) cot(c)
• Limx-gtc sec(x) sec(c)
• Limx-gtc csc(x) csc(c)

7
Another Limit Theorem

• If F(X) G(X) for all x? c in an open interval
  containing c , then if both F(x) and G(X) have a
  limit as x approaches c, then
  Limx-gtcF(x) Limx-gtc G(x).

8
The Squeeze Theorem

• If H(x) lt F(x) lt G(x) for all x in an open
  interval containing c, except possibly at c
  itself, and if Limx-gtc H(x) Limx-gtc G(x) L,
  then Limx-gtc F(x) L.

9
Two Important Trigonometric Limits

• Limx-gt0 sin(x)/x 1
• Limx-gt0 (1 cos(x)/x 0