Identities

Question 1

lim{x→a}[ f(x) + g(x) ]

Answer 1

lim{x→a}[ f(x) ] + lim{x→a}[ g(x) ]

Question 2

lim{x→a}[ f(x) - g(x) ]

Answer 2

lim{x→a}[ f(x) ] - lim{x→a}[ g(x) ]

Question 3

lim{x→a}[ c * f(x) ]

Answer 3

c * lim{x→a}[ f(x) ]

Question 4

lim{x→a}[ f(x) * g(x) ]

Answer 4

lim{x→a}[ f(x) ] * lim{x→a}[ g(x) ]

Question 5

lim{x→a}[ (f(x)) ÷ (g(x)) ]

Answer 5

( lim{x→a}[ f(x) ] ) ÷ ( lim{x→a}[ g(x) ] )

Question 6

lim{x→a}[ c ]

Answer 6

c

Question 7

lim{x→a}[ ⁿ√[ f(x) ] ]

Answer 7

ⁿ√[ lim{x→a}[ f(x) ] ]

Question 8

If g(x) ≤ f(x) ≤ h(x) when x≅a (except possibly when x=a), and lim{x→a}[ g(x) ] = lim{x→a}[ h(x) ], then…

Answer 8

lim{x→a}[ g(x) ] = lim{x→a}[ f(x) ] = lim{x→a}[ h(x) ]

Question 9

lim{t→0}[ sin(t)/t ]

Answer 9

1

Question 10

lim{t→0}[ (cos(t) - 1)/t ]

Answer 10

0

Question 11

[f(x) + g(x)]’

Answer 11

f’(x) + g’(x)

Question 12

[f(x) - g(x)]’

Answer 12

f’(x) - g’(x)

Question 13

[f(x) * g(x)]’

Answer 13

f(x) * g’(x) + g(x) * f’(x)

Question 14

[f(x) / g(x)]’

Answer 14

[g(x) * f’(x) - f(x) * g’(x)] / [g(x)]^2

Question 15

[a * f(x)]’

Answer 15

a * f’(x)

Question 16

[x^n]’

Answer 16

nx^(n-1)

Question 17

d/dx [c]

Answer 17

0

Question 18

[sin(x)]’

Answer 18

cos(x)

Question 19

[cos(x)]’

Answer 19

-sin(x)

Question 20

[tan(x)]’

Answer 20

[sec(x)]^2

Question 21

[cot(x)]’

Answer 21

-[csc(x)]^2

Question 22

[sec(x)]’

Answer 22

sec(x) * tan(x)

Question 23

[csc(x)]’

Answer 23

-csc(x) * cot(x)

Question 24

∫[x^n]

Answer 24

(x^(n+1))/(n+1)

Question 25

{i=1, n}Σ{1}

Answer 25

n

Question 26

{i=1, n}Σ{i}

Answer 26

n(n+1)/2

Question 27

{i=1, n}Σ{i^2}

Answer 27

n(n+1)(2n+1)/6

Question 28

{i=1, n}Σ{i^3}

Answer 28

(n)(n)(n+1)(n+1)/4

Question 29

log(a, (a^x))

Answer 29

x

Question 30

a^(log(a, x))

Answer 30

x

Question 31

lim{t→0}{(x^t - 1)/t}

Answer 31

ln(x)

Question 32

d/dx{e^x}

Answer 32

e^x

Question 33

d/dx{a^x}

Answer 33

a^x ⋅ ln(a)

Question 34

{ ∫ {1/x} dx}

Answer 34

ln(|x|) + c

Question 35

d/dx {ln(u(x))}

Answer 35

u'(x)/u(x)

Question 36

d/dx {log(a, u(x))}

Answer 36

u'(x)/( ln(a) ⋅ u(x) )

Question 37

log(b, x) + log(b, y)

Answer 37

log(b, (xy))

Question 38

log(b, x) - log(b, y)

Answer 38

log(b, (x/y))

Question 39

lim{x→0+}{ln(x)}

Answer 39

-∞

Question 40

(log(b, x))/(log(b, y))

Answer 40

log(y, x)

Question 41

(sin(x))^2 + (cos(x))^2

Answer 41

1

Question 42

(tan(x))^2 + 1

Answer 42

(sec(x))^2

Question 43

(cot(x))^2 + 1

Answer 43

(csc(x))^2

Question 44

1/sin(x)

Answer 44

csc(x)

Question 45

1/cos(x)

Answer 45

sec(x)

Question 46

1/tan(x)

Answer 46

cot(x)