Mathematics 53 2nd Semester, A.Y. 2014-2015Exercises 2 - Limits II Q3, R3, W8, X8

I. Evaluate the following limits, if they exist.

1. limx4+

3x2 + 5x+ 2

x2 2x 8

2. limx1

x+ x2 + x3

1 x3

3. limx4+

2x2 7x 4|x 4|

4. limw0

(1

w 1|w|

)

5. limx6

x 6[[x 6]]

6. lims1+

[[s]] + [[2s]] + 1s

s2 2

7. limy1+

5 y2 + [[y]][[y + 3]] y

8. limt1

[[t]] 2|t+ 1| 5

9. limx3

|x 3|2x [[x+ 4]]

10. limx1

x 1x2 2x 3

11. limx2

(1

x 2 +1

x2 5x+ 6)

12. limr2+

(1

r2 4 +1

r + 2

)

13. limx1+

(2x

x 1 2

x2 + 4x 5)

14. limu2

(1

|u 2| +1

u2 4)

15. limx1/3

(1

6x2 + 2x 2|3x+ 1|

)II. Consider the following piecewise functions and do as indicated.

1. Let f(x) =

x2 2x 3

x+ 1, if x < 0

[[x+ 1]] , if 0 x < 2|2x| 2, if x 2

.

Evaluate each of the following, if its value exists.

(a) limx1

f(x) (b) limx0

f(x) (c) limx1

f(x) (d) limx2

f(x)

2. Let g(x) =

x2 x 6x+ 6

, if x < 0

[[2x 1]] , if 0 < x < 1|x 5|2 x if x 1

.

Evaluate each of the following, if its value exists.

(a) limx6

g(x) (b) limx0

g(x) (c) limx1

g(x) (d) limx2

g(x)

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