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Question 1

lim x->0 sinx/x =

Answer 1

= 1

Question 2

lim x->0 1-cosx/x² =

Answer 2

= 1/2

Question 3

lim x->∞ (1+1/x)² =

Answer 3

= e

Question 4

lim x-> -∞ (1+1/x)²=

Answer 4

= e

Question 5

lim x->0 log(1+x)/x =

Answer 5

= 1

Question 6

lim x-> 0 eˣ -1 / x =

Answer 6

= 1

Question 7

lim x->∞ eˣ/xᵇ

Answer 7

= ∞

Question 8

lim x->∞ logx/xᵇ =

Answer 8

= 0

Question 9

lim x->0+ xᵇlogx =

Answer 9

= 0

Question 10

d/dx logx

Answer 10

1/x

Question 11

d/dx sinx

Answer 11

cosx

Question 12

d/dx cosx

Answer 12

-sinx

Question 13

d/dx tanx

Answer 13

sec²x

Question 14

d/dx cotx

Answer 14

-cosec²x

Question 15

d/dx arcsinx

Answer 15

1/√(1-x²)

Question 16

d/dx arccosx

Answer 16

-1/√(1-x²)

Question 17

d/dx arctanx

Answer 17

1/√(1+x²)

Question 18

d/dx sinhx

Answer 18

coshx

Question 19

d/dx coshx

Answer 19

sinhx

Question 20

d/ dx arccoshx

Answer 20

1/√(x²-1)

Question 21

Sequence Convergence Tests

Answer 21

1. Null sequence test
2. The comparison test
3. The ratio test
4. The root test
5. The Integral test
6. Alternating Series Test
7. Absolute Convergence Test

Question 22

Null Sequence Test

Answer 22

if Σaₙ converges, then an->0

test for divergence only: if an->0 then Σaₙ diverges

Question 23

The Comparison Test

Answer 23

Suppose an>=0 bn>=an for n then:

1. Σbₙ converges <=> Σaₙ converges
2. Σaₙ diverges <=> Σbₙ diverges

Question 24

The Ratio Test

Answer 24

Suppose Σaₙ is a series of non-negative terms and:
aₙ₊₁/aₙ -> r
then: 
1. if r<1 => Σaₙ converges
2. if r>1 Σaₙ  diverges
3. if r=0 inconclusive

Question 25

The root test

Answer 25

Suppose Σaₙ is a series of non-negative terms and aₙ¹/ⁿ->. Then:

1. if r<1 => Σaₙ converges
2. if r>1 => Σaₙ diverges
3. if r=0 inconclusive

Question 26

Integral Test

Answer 26

Suppose f:[1,∞)-> is continuous, decreasing and positive in its domain. Then the series Σfₙ converges if and only if the sequence (∫f(x)dx) converges

Question 27

Alternating series test

Answer 27

Consider the series Σ(-1)ⁿ⁺¹aₙ where the sequence (aₙ) is decreasing and converging to zero. Then the series converges.

Question 28

Absolute Convergence Test

Answer 28

If a series converges absolutely absolutely, then it converges