1st - x → -4-, x <-4, so x + 4 < 0. lim x →-4-| x...

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

View Full Document Right Arrow Icon
Calculus With Analytic Geometry Hakan TOR February 26,2010 1. Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downwards. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3. Solution: The answers are below, but for explanations they are not enough. (a) f(x)+3 (b) f(x)-3 (c) f(x-3) (d) f(x+3) (e) -f(x) (f) f(-x) (g) 3f(x) (h) f(x)/3 2. (a) What is wrong with the following equation? x 2 + x - 6 x - 2 = x + 3 (b) In view of part (a), explain why the equation lim x 2 x 2 + x - 6 x - 2 = lim x 2 ( x + 3) is correct. Solution: 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
3. Use the Squeeze Theorem to show that lim x 0 p x 3 + x 2 sin π x = 0 Solution: Observe - p x 3 + x 2 p x 3 + x 2 sin π x p x 3 + x 2 since - 1 sin π x 1 x IR Since lim x 0 x 3 + x 2 = 0, lim x 0 x 3 + x 2 sin π x = 0 by the Squeeze Theorem. 4. Find lim x →- 4 - | x + 4 | x + 4 . Solution: When
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x → -4-, x <-4, so x + 4 < 0. lim x →-4-| x + 4 | x + 4 = lim x →-4--( x + 4) x + 4 = lim x →-4--1 =-1 5. *Prove that lim x → 2 1 x = 1 2 . Solution: 6. *If the function f is defined by ± if x is rational 1 if x is irrational prove that lim x → f ( x ) does not exist. Solution: 7. Suppose that lim x → a f ( x ) = ∞ and lim x → a g ( x ) = c , where c is a real number. Prove each statement. (a) lim x → a [ f ( x ) + g ( x )] = ∞ (b) lim x → a [ f ( x ) g ( x )] = ∞ if c > (c) lim x → a [ f ( x ) g ( x )] =-∞ if c < Solution: 8. From the graph of g, state the intervals on which g is continuous. Figure 1: Graph Solution: 9. Use the Intermediate Value Theorem to prove that there is a positive number c such that c 2 = 0. (This proves existence of the number √ 2) Solution: 2...
View Full Document

This note was uploaded on 04/30/2010 for the course MATHEMATIC MATH 119 taught by Professor Tor during the Spring '10 term at Middle East Technical University.

Page1 / 2

1st - x → -4-, x <-4, so x + 4 < 0. lim x →-4-| x...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online