University:
University of California, Berkeley
Course:
MATH 56 Linear Algebra
Academic year:

2024
Views:

95

Pages:

1
Author:

Clint W.

De nition: Let f be a function de ned on some interval (a; 1). Then limx!1 f (x) = L means that the values of f (x) can be made arbitrarily close to L by taking x suciently large. Exercise 1. Find limx!1 x1 . De ntion: Let f be a function de ned on some interval ( limx! 1 f (x) = 1 ). Then ;a L means that the value of f (x) can be made arbitrarily close to negative. L by taking x suciently large Exercise 2. Find limx! 1 x1 . Exercise 3. Sketch the graph of an example of a function f that satis es all of the given conditions: limx!2 f (x) = 1, limx! 2+ f (x) = 1, limx! 2 f (x) = 1, limx! 1 f (x) = 0, limx!1 f (x) = 0, f (0) = 0. (#6) Class Exercise 1. Sketch the graph of an example of a function f that satis es all of the given conditions: limx!1 f (x) = 3, limx!2 f (x) = 1, limx!2+ f (x) = 1, f is odd. (#8) Class Exercise 2. Sketch the graph of an example of a function f that satis es all of the given conditions: limx!3 f (x) = 1, limx!1 f (x) = 2, f (0) = 0, f is even. (#10) Exercise 4. Find limx!1 (5 + x1 ). (Hass Sec 2.6 Ex 2a) Exercise 5. Find limx! 1 p 3 . (Hass Sec 2.6 Ex 2b) x2 Exercise 6. Find the limit or show that it does not exist. (#16, 26) p 22, 2 2 1 x x 2 p (a) limx!1 x3 x+1 (b) limx!1 x4 +1 (c) limx! 1 x + x + 2x Class Exercise 3. Find the limit or show that (#18,20,24,28,30,32,34) p it does not exist. p 3 2 t t 9x6 x (d) limx!1 px2 + 1 (a) limx! 1 42xx3+64xx+52 (b) limt!1 2t3t=2 +3 (c) lim x! 1 x3 +1 t 5 3x 3x x6 (e) limx!1 (e x + 2 cos 3x) (f) limx! 1 1+ (g) limx!1 ee3x +ee 3x x4 +1 De nition: A line y = b is a horizontal asymptote of the graph of a function y = f (x) if either limx!1 f (x) = b or limx! 1 f (x) = b. 3 Exercise 7. Find the horizontal asymptotes of the graph of f (x) = jxxj3 +12 . (Hass Sec 2.6 Ex 4) Class Exercise 4. Find the horizontal and vertical asymptotes of each curve. (#42, 44, 46) 2 1+x4 2ex (a) y = 2x2x +1 3x 3 (b) y = x2 x4 (c) y = ex 5 1