Download 2.1-2.2 Review - Perry Local Schools

2.1-2.2 Review – Calculus
Honors Precalculus – Meinke
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Definition: 𝑓(𝑥) = 𝑖𝑛𝑡𝑥 is called the “greatest integer function”. It is also called the “rounding down
function”.
Examples: 𝑖𝑛𝑡2 = _____
5
𝑖𝑛𝑡 2 = _____
5
1
𝑖𝑛𝑡(− 2) = _____
𝑖𝑛𝑡 2 = _____
Evaluate the following limits.
Direct substitution:
1.) lim 2
𝑥→3
4.) lim
3.) lim 3−2𝑥
5.) lim 𝑖𝑛𝑡(−𝑥 − 0.5)
6.) lim [𝑖𝑛𝑡(𝑥 + 1) − 𝑖𝑛𝑡𝑥 2 ]
𝑥→5−
𝑥 2 +4
𝑥→−2 𝑥−2
6𝑥−1
2.) lim 𝜋𝑥
𝑥→−3
𝑥→0
𝑥→3−
Algebraic Simplification, Then Substitution:
7.) lim
𝑥 2 −4
8.) lim
𝑥→2 𝑥−2
10.) lim
𝑥→−𝑎
2𝑥 2 −𝑥−15
𝑥→3
𝑥 2 +2𝑎𝑥+𝑎2
𝑥+𝑎
𝑥−3
𝑥 2 −2𝑥
11.) lim 𝑥 2 +𝑥−6
𝑥→2
9.) lim
𝑥+7
𝑥→−7+ 𝑥 2 −49
12.) lim
𝑠𝑖𝑛5𝑥
𝑥→0 2𝑥
Limits at Infinity:
−5
13.) lim
𝑥→∞
14.) lim
𝑥2
𝑥→∞
15.) lim
𝑥→−∞ 5
6𝑥 3 +4𝑥
16.) lim
𝑥2
−𝑥 3 −4𝑥 2 +5
19.) lim 𝑡𝑎𝑛𝑥
𝑥→−∞
𝑥→∞
𝑥 5 −1
17.) lim
𝑥→−∞ 66𝑥 2 +𝑥−1
20.) lim
𝑥→∞
1
1+𝑒 −𝑥
2𝑥 3 +𝑥 2 +2
𝑥 4 −75
18.) lim 𝑠𝑖𝑛𝑥
𝑥→∞
21.) lim
1
𝑥→−∞ 1+𝑒 −𝑥
−1, 𝑥 < 0
Definition: 𝑠𝑔𝑛𝑥 = { 0, 𝑥 = 0
1, 𝑥 > 0
22.) lim
𝑥2
𝑥→−∞ 𝑠𝑔𝑛𝑥
23.) lim 2𝑠𝑔𝑛𝑥
𝑥→∞
24.) lim
𝑥3
𝑥→−∞ 𝑠𝑔𝑛𝑥
Limits at V.A:
25.) lim
𝑥 2 +4
𝑥→2 𝑥−2
𝑥 2 −4
28.) lim (𝑥−2)3
𝑥→2
𝑥 2 +4
26.) lim (𝑥−2)2
𝑥→2
4−𝑥 2
29.) lim (𝑥−2)3
𝑥→2
27.) lim
𝑥 2 +4
𝑥→−2 𝑥+2
𝑥 2 −4
30.) lim (𝑥−2)2
𝑥→2
1
1
31.) lim (𝑥 − 𝑥 2 )
𝑥→0+
−1
32.) lim
𝑥→2− 2−𝑥
1
33.) lim (𝑥−3)2
𝑥→3
Other Limits:
34.) lim 𝑖𝑛𝑡𝑥
35.) lim 𝑖𝑛𝑡𝑥
𝑥→4
37.) lim
𝑥→4.5
|𝑥|
38.) lim
𝑥→0 𝑥
|𝑥|
36.) lim 𝑖𝑛𝑡𝑥
𝑥→−4
39.) lim
𝑥→0− 𝑥
|𝑥|
𝑥→0+ 𝑥
Definition: If lim 𝑓(𝑥) = 𝐿 and/or lim 𝑓(𝑥) = 𝑀, where L and M are finite, then 𝑓(𝑥) has a H.A. at
𝑥→∞
𝑥→−∞
𝑦 = 𝐿 and/or 𝑦 = 𝑀.
Find the horizontal asymptote(s).
40.) 𝑦 =
1
𝑥
1+𝑐𝑜𝑠( )
1
1+
𝑥
1
43.) 𝑦 = 1+𝑒 𝑥
41.) 𝑦 =
44.) 𝑦 =
2𝑥+𝑠𝑖𝑛𝑥
𝑥
−3𝑥+1
2𝑥+2
𝑠𝑖𝑛𝑥
42.) 𝑦 = 2𝑥 2 +𝑥
45.) 𝑦 = 𝑠𝑒𝑐𝑥
46.) Let 𝑓(𝑥) = {
−𝑥 2 − 1, 𝑥 < 1
.
𝑥 − 1, 𝑥 ≥ 1
Find the following:
𝑎. ) lim 𝑓(𝑥)
𝑥→1−
b.) lim 𝑓(𝑥)
c.) lim 𝑓(𝑥)
𝑥→1+
𝑥→1
47.) Given the graph of g(x) below, find the following:
a.) lim 𝑔(𝑥)
𝑥→0
b.) lim 𝑔(𝑥)
𝑥→∞
c.) lim 𝑔(𝑥)
𝑥→−∞
d.) lim 𝑔(𝑥)
𝑥→1+
48.) Sketch a function p(x) with the following conditions:
a.) p(x) has a limit at x = -2.
b.) p(-2) is undefined.
c.) lim 𝑝(𝑥) = 2
𝑥→∞
d.) lim 𝑝(𝑥) = 2
𝑥→−∞
e.) lim 𝑝(𝑥) = 1
𝑥→1−
f.) lim 𝑝(𝑥) = 3
𝑥→1+
g.) p(1) = -1
h.) lim 𝑝(𝑥) = −∞
𝑥→−1
Find the right and left-end behavior models.
49.) 𝑓(𝑥) = −𝑒 −𝑥 + 𝑥 8
50.) 𝑔(𝑥) = 𝑥 + 𝑠𝑖𝑛𝑥