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MAST10006 Calculus 2 Semester 2, 2022
Question 1 (13 marks)
(a) Evaluate the limit lim
θ→0
cos MAST10006 Calculus 2 Formulae Sheet
Z
sin x dx = ? cos x + C
Z
cos x dx = sin x + C
Z
sec x dx = log |sec x + tan x| + C
Z
cosec x dx = log |cosec x ? cot x| + C
Z
sec2 x dx = tan x + C
Z
cosec 2x dx = ? cot x + C
Z
sinh x dx = cosh x + C
Z
cosh x dx = sinh x + C
Z
sech 2
x dx = tanh x + C
Z
cosech 2
x dx = ? coth x + C
Z
1
√
a
2 ? x
2
dx = arcsin x
a

+ C
Z
1
√
x
2 + a
2
dx = arcsinh x
a

+ C
Z
?1
√
a
2 ? x
2
dx = arccos x
a

+ C
Z
1
√
x
2 ? a
2
dx = arccosh x
a

+ C
Z
1
a
2 + x
2
dx =
1
a
arctan x
a

+ C
Z
1
a
2 ? x
2
dx =
1
a
arctanh x
a

+ C
where a > 0 is constant and C is an arbitrary constant of integration.
cos2 x + sin2 x = 1 cosh2
x ? sinh2
x = 1
1 + tan2 x = sec2 x 1 ? tanh2
x = sech 2
x
cot2 x + 1 = cosec 2x coth2
x ? 1 = cosech 2
x
cos(2x) = cos2 x ? sin2 x cosh(2x) = cosh2
x + sinh2
x
cos(2x) = 2 cos2 x ? 1 cosh(2x) = 2 cosh2
x ? 1
cos(2x) = 1 ? 2 sin2 x cosh(2x) = 1 + 2 sinh2
x
sin(2x) = 2 sin x cos x sinh(2x) = 2 sinh x cosh x
cos(x + y) = cos x cos y ? sin x sin y cosh(x + y) = cosh x cosh y + sinh x sinh y
sin(x + y) = sin x cos y + cos x sin y sinh(x + y) = sinh x cosh y + cosh x sinh y
cosh x =
MAST10006 Calculus 2 Formulae Sheet
Z
sin x dx = ? cos x + C
Z
cos x dx = sin x + C
Z
sec x dx = log |sec x + tan x| + C
Z
cosec x dx = log |cosec x ? cot x| + C
Z
sec2 x dx = tan x + C
Z
cosec 2x dx = ? cot x + C
Z
sinh x dx = cosh x + C
Z
cosh x dx = sinh x + C
Z
sech 2
x dx = tanh x + C
Z
cosech 2
x dx = ? coth x + C
Z
1
√
a
2 ? x
2
dx = arcsin x
a

+ C
Z
1
√
x
2 + a
2
dx = arcsinh x
a

+ C
Z
?1
√
a
2 ? x
2
dx = arccos x
a

+ C
Z
1
√
x
2 ? a
2
dx = arccosh x
a

+ C
Z
1
a
2 + x
2
dx =
1
a
arctan x
a

+ C
Z
1
a
2 ? x
2
dx =
1
a
arctanh x
a

+ C
where a > 0 is constant and C is an arbitrary constant of integration.
cos2 x + sin2 x = 1 cosh2
x ? sinh2
x = 1
1 + tan2 x = sec2 x 1 ? tanh2
x = sech 2
x
cot2 x + 1 = cosec 2x coth2
x ? 1 = cosech 2
x
cos(2x) = cos2 x ? sin2 x cosh(2x) = cosh2
x + sinh2
x
cos(2x) = 2 cos2 x ? 1 cosh(2x) = 2 cosh2
x ? 1
cos(2x) = 1 ? 2 sin2 x cosh(2x) = 1 + 2 sinh2
x
sin(2x) = 2 sin x cos x sinh(2x) = 2 sinh x cosh x
cos(x + y) = cos x cos y ? sin x sin y cosh(x + y) = cosh x cosh y + sinh x sinh y
sin(x + y) = sin x cos y + cos x sin y sinh(x + y) = sinh x cosh y + cosh x sinh y
cosh x =
1
2