AP Calculus AB
Name ____________________________________
Limits and Continuity Review
Date ________________________ Pd. _________
1. If f ( x) =
x2 − 9
is continuous at x = -3, then f (-3) = ____________________
x+3
(3 − x )
lim
x →0 ( x − 3 )
2
2.
=
3. lim(sin 2 x)
x →π
⎧ x 2 + bx, x ≤ 5
⎪
4. Non-Calculator If is a continuous function defined by f ( x) = ⎨
⎛πx ⎞
⎪5sin ⎜ 2 ⎟ , x > 5
⎝
⎠
⎩
Find b.
5. If y = 7 is a horizontal asymptote of a rational function f, then which of the following is true
a) lim f ( x) = ∞
x→7
b) lim f ( x) = 7
x →∞
d) lim f ( x) = 0
x →7
c) lim f ( x) = 7
x →0
e) lim f ( x) = −7
x→−∞
6. Non-Calculator Which of the following is continuous at x = 0?
II. f ( x) = e x
I. f ( x) = x
a) I only
b) II only
III. f ( x) = ln(e x − 1)
c) I and II only
d) II and III only
⎧ln x, 0 < x ≤ 2
7. Non-Calculator If f ( x) = ⎨ 2
then lim f ( x) is
x →2
⎩ x ln 2, 2 < x ≤ 4
a) ln 2
b) ln 8
c) ln 16
d) 4
e) nonexsistent
e) none
⎧ −x, x < 0
⎪
8. Let f (x) = ⎨3 − x, 0 ≤ x < 3
⎪( x − 3) 2 , x > 3
⎩
a) lim+ f ( x) =
b) lim− f ( x) =
c) lim f ( x) =
d) lim− f ( x) =
e) lim+ f ( x) =
f) lim f ( x) =
x →0
x →0
x→0
x →3
x →3
x →3
⎧⎪ x + 1, 0 ≤ x ≤ 3
9. Let the function f be defined by f ( x) = ⎨
⎪⎩5 − x, 3 < x ≤ 5
Is f(x) continuous at x = 3? Why or why not?
10. lim+
x →0
ln x
=
x
sin x
=
x →∞ e + cos x
11. lim
12. Let f be the function given by f(x) = 2 xe2 x
a) Find lim f ( x)
x →−∞
and
lim f ( x)
x→∞
x