James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

166 Chapter 2 Limits and Derivatives
15. (a) What does it mean for f to be differentiable at a?
(b) What is the relation between the differentiability and
continuity of a function?
(c) Sketch the graph of a function that is continuous but not
differentiable at a − 2.
16. Describe several ways in which a function can fail to be
differentiable. Illustrate with sketches.
TRUE-FALSE QUIZ
Determine whether the statement is true or false. If it is true,
explain why. If it is false, explain why or give an example that
disproves the statement.
1. lim
x l4S 2x
x 2 4 2 8
x 2 4D − lim
x l4
2x
x 2 4 2 lim
x l4
x 2 lim
1 6x 2 7
sx 2 1 6x 2 7d
2. lim
x l1 x 2 1 5x 2 6 − x l1
lim sx 2 1 5x 2 6d
x l1
lim sx 2 3d
x 2 3
3. lim
x l 1 x 2 1 2x 2 4 −
x l 1
lim sx 2 1 2x 2 4d
x l 1
4. x 2 2 9
x 2 3 − x 1 3
x 2 2 9
5. lim
x l 3 x 2 3 − lim sx 1 3d
x l 3
6. If lim x l 5 f sxd − 2 and lim x l 5 tsxd − 0, then
lim x l 5 f f sxdytsxdg does not exist.
7. If lim x l5 f sxd − 0 and lim x l 5 tsxd − 0, then
lim x l 5 f f sxdytsxdg does not exist.
8. If neither lim x l a f sxd nor lim x l a tsxd exists, then
lim x l a f f sxd 1 tsxdg does not exist.
8
x 2 4
9. If lim x l a f sxd exists but lim x l a tsxd does not exist, then
lim x l a f f sxd 1 tsxdg does not exist.
10. If lim x l 6 f f sxd tsxdg exists, then the limit must be f s6d ts6d.
11. If p is a polynomial, then lim x l b psxd − psbd.
12. If lim x l 0 f sxd − ` and lim x l 0 tsxd − `, then
lim x l 0 f f sxd 2 tsxdg − 0.
13. A function can have two different horizontal asymptotes.
14. If f has domain f0, `d and has no horizontal asymptote, then
lim x l ` f sxd − ` or lim x l ` f sxd − 2`.
15. If the line x − 1 is a vertical asymptote of y − f sxd, then f is
not defined at 1.
16. If f s1d . 0 and f s3d , 0, then there exists a number c
between 1 and 3 such that f scd − 0.
17. If f is continuous at 5 and f s5d − 2 and f s4d − 3, then
lim x l 2 f s4x 2 2 11d − 2.
18. If f is continuous on f21, 1g and f s21d − 4 and f s1d − 3,
then there exists a number r such that | r | , 1 and f srd − .
19. Let f be a function such that lim x l 0 f sxd − 6. Then there
exists a positive number such that if 0 , | x |
| f sxd 2 6 | , 1.
20. If f sxd . 1 for all x and lim x l 0 f sxd exists, then
lim x l 0 f sxd . 1.
21. If f is continuous at a, then f is differentiable at a.
22. If f 9srd exists, then lim x l r f sxd − f srd.
d 2 y
23. S
dx − dy
2
dxD
2
24. The equation x 10 2 10x 2 1 5 − 0 has a root in the
interval s0, 2d.
25. If f is continuous at a, so is | f | .
26. If | f | is continuous at a, so is f .
, , then
EXERCISES
1. The graph of f is given.
y
1
0 1
x
(a) Find each limit, or explain why it does not exist.
(i) lim f sxd (ii) lim f sxd
x l21 x l23 1
(iii) lim f sxd
x l23
(iv) lim f sxd (v) lim f sxd (vi) lim f sxd
x l4 x l0 x l2 2
(vii) lim f sxd (viii) lim f sxd
x l` x l 2`
(b) State the equations of the horizontal asymptotes.
(c) State the equations of the vertical asymptotes.
(d) At what numbers is f discontinuous? Explain.
2. Sketch the graph of a function f that satisfies all of the
following conditions:
lim f sxd − 22, lim f sxd − 0, lim
x l2` x l `
lim f sxd − 2`, lim
x l32 f sxd − 2,
x l31 f is continuous from the right at 3
f sxd − `,
x l23
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