# 0112 - Math 31A 2010.01.12 MATH 31A DISCUSSION JED YANG 1...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 31A 2010.01.12 MATH 31A DISCUSSION JED YANG 1. Limits 1.1. Basic Limit Laws. Assume that lim x → c f ( x ) and lim x → c g ( x ) exist. Then: (a) Sum Law: lim x → c ( f ( x ) + g ( x )) = lim x → c f ( x ) + lim x → c g ( x ) . (b) Constant Multiple Law: For any number k ∈ R , lim x → c kf ( x ) = k lim x → c f ( x ) . (c) Product Law: lim x → c ( f ( x ) g ( x )) = parenleftBig lim x → c f ( x ) parenrightBigparenleftBig lim x → c g ( x ) parenrightBig . (d) Quotient Law: If lim x → c g ( x ) negationslash = 0, then lim x → c f ( x ) g ( x ) = lim x → c f ( x ) lim x → c g ( x ) . 1.2. Exercise 2.3.22. Evaluate the limit lim z → 1 z- 1 + z z +1 . Solution. Recall that lim z → 1 z = 1 and lim z → 1 1 = 1. By the Quotient Law, lim z → 1 z- 1 = lim z → 1 1 lim z → 1 z = 1 1 = 1. By the Sum Law, lim z → 1 z- 1 + z = lim z → 1 z- 1 + lim z → 1 z = 1+1 = 2. By the Sum Law, lim z → 1 z +1 = 2. So by the Quotient Law, lim z → 1 z- 1 + z z +1 = lim z → 1 z- 1 + z lim z → 1 z +1 = 2 2 = 1. square 1.3. Exercise 2.3.29. Can the Quotient Law be applied to evaluate lim x → sin x x ? Solution. The Quotient Law requires the limit of the denominator, namely, lim x → x , to exist and be nonzero. This is not the case, so we cannot apply directly. square 1.4. Exercise 2.3.30. Show that the Product Law cannot be used to evaluate lim x → π/ 2 ( x − π/ 2) tan x . Solution. The Product Law requires the limit of each factor to exist. However, lim x → π/ 2 tan x does not exist. square 1.5. Exercise 2.3.31. Give an example where lim x → ( f ( x ) + g ( x )) exists but nei-...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

0112 - Math 31A 2010.01.12 MATH 31A DISCUSSION JED YANG 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online