103 worksheet limits PDF

Preview

Summary

Download 103 worksheet limits PDF

Description

201-103-RE - Calculus 1 WORKSHEET: LIMITS 1. Use the graph of the function f (x) to answer each question. Use ∞, −∞ or DN E where appropriate. (a)

f (0) =

(b)

f (2) =

(c)

f (3) =

(d) (e) (f) (g) (h)

lim f (x) =

x→0−

lim f (x) =

x→0

lim f (x) =

x→3+

lim f (x) =

x→3

lim f (x) =

x→−∞

2. Use the graph of the function f (x) to answer each question. Use ∞, −∞ or DN E where appropriate.

(a)

f (0) =

(b)

f (2) =

(c)

f (3) =

(d) (e) (f) (g)

lim f (x) =

x→−1

lim f (x) =

x→0

lim f (x) =

x→2+

lim f (x) =

x→∞

3. Evaluate each limit using algebraic techniques. Use ∞, −∞ or DN E where appropriate. (a) (b)

lim

x→0

lim

x→5

x2 − 25 x2 − 4x − 5

(q)

x2 − 25 x2 − 4x − 5

(r)

2

(c) (d) (e) (f) (g) (h)

(i) (j)

(k)

(l) (m) (n) (o) (p)

lim

x→1

7x − 4x − 3 3x2 − 4x + 1

(s)

x4 + 5x3 + 6x2 lim 2 x→−2 x (x + 1) − 4(x + 1) lim |x + 1| +

x→−3

lim

√

x→3

lim

√

x→3

3 x

(u)

x+1−2 x2 − 9

(v)

x2 + 7 − 3 x+3

(w)

x2 + 2x − 8

lim √

x2 + 5 − (x + 1)  2  2y + 2y + 4 1/3 lim y→5 6y − 3 p lim 4 2 cos(x) − 5

x→2

x→0

1 1 − lim 3 + x 3 − x x→0 x 1 2x + 8 − 2 x lim x − 12 x→−6 x+6 √ √ lim x2 − 2 − x2 + 1 x→∞

lim

x→−∞

lim

x→7

lim−

x→1

√

√ 6

(t)

x−2−

2x − 14

√

3 − 3x

√

(x) (y) (z) (A) (B) (C) (D)

x (E) (F)

x4 − 10 x→∞ 4x3 + x r x−3 lim 3 x→−∞ 5−x lim

3x3 + x2 − 2 x2 + x − 2x3 + 1

lim

x→∞

x+5 2x2 + 1   x5 + 1 lim cos 6 x→−∞ x + x5 + 100 lim

x→∞

2x −4

lim

x→2 x2

lim

x→−1

x2

3x + 2x + 1

x2 − 25 x→−1 x2 − 4x − 5 √ x2 − 5 + 2 lim x→3 x−3 lim

2x + sin(x) x→0 x4 1 2 lim + ex x→1− x − 1 lim

lim 2x2 − 3x

x→∞

√

√ x+2− 2−x lim x→0 x x e lim + 1 + ln(x) x→0 √ lim x2 + 1 − 2x x→∞

√ 3 x−1 lim √ x→1 x−1

4. Find the following limits involving absolute values. (a) lim

x→1

x2 − 1 |x − 1|

(b) lim

x→−2

1 + x2 |x + 2|

(c) lim

x→3−

x2 |x − 3| x−3

5. Find the value of the parameter k to make the following limit exist and be finite. What is then the value of the limit? x2 + kx − 20 lim x→5 x−5 6. Answer the following questions for the piecewise defined function f (x) described on the right hand side. (a)

f (1) =

(b)

lim f (x) =

(c)

x→0

lim f (x) =

f (x) =

(

sin(πx) 2

2x

for x < 1, for x > 1.

x→1

7. Answer the following questions for the piecewise defined function f (t) described on the right hand side. (a)

f (−3/2) =

(b)

f (2) =

(c)

f (3/2) =

(d) (e) (f) (g)

lim f (t) =

t→−2

lim+ f (t) =

t→−1

lim f (t) =

t→2

lim f (t) =

t→0

 t2     t+6 f (t) =  t2 − t    3t − 2

for t < −2 for − 1 < t < 2 for t ≥ 2

ANSWERS: 1. (a) DNE 2. (a) 0

(b) 0

(c) 3

(d) −∞

(b) DNE

(c) 0

(d) DNE

(e) DNE (e) 0

(f) 2 (f) −∞

(g) DNE

(h) 1

(g) 1

3. (a) 5 (b)

(l)

5 3

1 36

(w) −∞

(m) 0

(x) DNE

(c) 5

(n) DNE

(d) 1

(o) DNE

(e) 1

(p) 0

(f) (g)

1 24 1 6

(A) −∞ (B) ∞

(r) −1 (s)

(t) 0

(j) DNE

(u) 1

(k) − 29

(v) DNE (b) ∞

(C)

− 32

4 3

4. (a) DNE

(z) ∞

(q) ∞

(h) −18 (i)

(y) DNE

(D) 0 (E) −∞ (F)

(c) −9

5. k = −1, limit is then equal to 9 6. (a) DNE

(b) 0

(c) DNE

7. (a) DNE

(b) 4

(c) 10

8. (a) 0

(b) 0

(c)

5 3

(d) DNE

√1 2

(e)

5 2

(f) 4

(g) DNE

2 3...