Fall 2014
Math 151
Week-in-Review 3
1. Given that
2. Find the infinite limit for the following:
𝑥−1
a. lim𝑥→0− 2
𝑥 (𝑥+2)
𝑥−1
b. lim𝑥→0+
𝑥 2 (𝑥+2)
𝑥−1
c. lim𝑥→0−
𝑥(𝑥+2)
𝑥−1
d. lim𝑥→0+
𝑥(𝑥+2)
3. What is wrong with the equation
𝑥 2 +𝑥−6
𝑥−2
=𝑥+3?
4. Evaluate each of the following limits if it exists:
2+𝑥 3 − 8
a. lim𝑥→0
b. lim𝑥→1
𝑥
𝑥 2 +𝑥−2
𝑥 2 −3𝑥+2
1
2
c. lim𝑥→1 [
d. lim𝑥→1.5
e.
−
𝑥−1
𝑥 2 −1
2
2𝑥 −3𝑥
|2𝑥−3|
lim𝑥→2− 𝑥 2 +
]
𝑥−2
5. The position of a moving particle at time t, 0 ≤ 𝑡 < 5 is given by the vector
2𝑡−10
𝑡−5
𝑟 𝑡 =
, 2
. What is the anticipated position of the particle at time t = 5?
𝑡−5
𝑡 −4𝑡−5
𝜋
6. Use the Squeeze theorem to show that lim𝑥→0 𝑥 3 + 𝑥 2 sin = 0
𝑥
7. Find the values of c and d that make the function f continuous everywhere
2𝑥,
𝑓 𝑥 = 𝑐𝑥 + 𝑑,
4𝑥,
2
𝑥<1
1≤𝑥≤2
𝑥>2
1
Fall 2014
Math 151
Week-in-Review 3
2
8. Given that
9. Which of the following functions f has a removable discontinuity at a?
10. If 𝑓 𝑥 = 𝑥 3 − 𝑥 2 − 10, show that a number c exists such that 𝑓 𝑐 = 10
11. Test the following functions for continuity. If not continuous, explain why.
a. 𝑓 𝑥 =
b. 𝑓 𝑥 =
𝑥 2 −1
, 𝑎 = −1
3,
𝑥=4
𝑥+1
𝑥 2 −2𝑥−8
𝑥−4
, 𝑥≠4
,𝑎 = 4
12. At what values of x is the following piece-wise function discontinuous?
5, 𝑥 ≤ −1
𝑓 𝑥 = −3𝑥, − 1 < 𝑥 ≤ 1
2𝑥
,
𝑥>1
𝑥−5
13. What are the horizontal and vertical asymptote(s) of the function 𝑓 𝑥 =
𝑥 2 −2𝑥−3
𝑥 4 −3𝑥 3 −4𝑥 2
?