Math Calculus Worksheet Chap 1: Limits and Their Properties Section: Name:

Mr. Lin

7

38 Theorem: The Squeeze Theorem If ℎ 𝑥 ≤ 𝑓(𝑥) ≤ 𝑔(𝑥) for all x in an open interval containing c, except possibly at c itself, and if lim!→! ℎ 𝑥 = 𝐿 = lim!→! 𝑔 𝑥 , then

_____________________ and ____________________.

39 Theorem: Two special trigonometric limits.

(1) Prove: lim!→!!"#!!= 1

Area of triangle ≥ Area of sector ≥ Area of triangle

___________ ≥  ____________ ≥  ______________. Multiply each expression by !

!"#! produces

___________ ≥  ____________ ≥  ______________.

Taking reciprocals and reversing the inequalities yields

___________ ≤  ____________ ≤  ______________.

Since cos(−𝜃) = ___________ and !"#(!!)(!!)

= _____________,

You can conclude that the inequality is valid for all

nonzero 𝜃 in the open interval ____________________.

Since lim!→! cos 𝜃 = ________ and lim!→! 1 = _________,

you can apply the Squeeze Theorem to conclude that

_____________________________________________.

(2) Prove: lim!→!!!!"#!

!= 0

40 Example: Find the limit: lim!→!!"#!!

.

41 Exercise: Find the limit: lim!→!!"#!!!

.

42 Example: Find the limit: lim!→!!"#!!!"#!!

.

43 Example: Find the limit: lim!→!!!!

!!!"#!!.

44 Example: Find the limit: lim!→!!!

!"#!!.

𝜃 𝜃 𝜃 1 1 1