MATH 171.501
Practice problems for Examination 1
For questions 1 to 6 circle the correct answer.
1. Given a = h2, −1i and b = h0, 3i, find a + 2b.
(a) h2, 2i
(b) h2, 5i
(c) h4, 5i
(d) h4, 1i
(e) h2, −7i
2. Which of the following vectors is orthogonal to h4, −5i?
(a) h3, −2i
(b) h3, 2i
(c) h5, 4i
(d) h4, −5i
(e) h5, −4i
2f (x)
.
x→a g(x)2 + 1
3. Given that lim f (x) = 3 and lim g(x) = 0, find lim
x→a
x→a
(a) 0
(b) 2
(c) 4
(d) 3
(e) 6
1
x4 + 3
4. Which of the following is a vertical asymptote for the curve y = 2
?
x −4
(a) x = 1
(b) x = 0
(c) x = 3
(d) x = −2
(e) x =
3
4
5. Which of the following is a horizontal asymptote for the curve y =
x2 + 1
?
3 − x2
(a) y = 0
(b) y = 1
(c) y = −1
(d) y =
1
3
(e) y = 3
6. In which of the following intervals does the equation x4 = 3 − x have a solution?
(a) (−1, 0)
(b) (0, 1)
(c) (1, 2)
(d) (2, 3)
(e) (3, 4)
2
7. (a) State what it means for the limit of a function f as x approaches a to be equal to
L.
(b) Give an ε-δ proof that lim (3x − 7) = −4.
x→1
8. Given a = h2, 3i and b = h1, 4i, find compa b and projb a.
9. Find all horizontal and vertical asymptotes for the curve y =
x2 + x − 2
.
x2 + 4x + 3
3x3 + x2 − 5
.
x→−∞
5x3 − 9
10. Evaluate lim
x3 − 1
.
x→1 x2 − 1
11. Evaluate lim
4x sin(1/x)
.
x→0 3 + sin(1/x)
12. Evaluate lim
13. (a) Define what it means for a function f to be continuous at a point a.
(b) Determine at which points the function


x2 + 3



 2
f (x) = x + 2


x−1



3x − 2
is continuous.
3
if x ≥ 1
if 0 < x < 1
if x ≤ 1