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(MATH023)[2009](f)midterm~dli^_10404.pdf
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Time Allowed: 2 Hours Total Marks: 100
1. (40 Marks) Find the following quantities:
n
x . 1
(a)
lim .
x1 x2 . 1
1 . sin x 2
(b)
lim .
x (x . )2
(c) lim [1+ xP (x)]1/x , where P (x) is a polynomial.
x0
arcsin x
(d) lim .
x0 sin x
(e) y., where xf(y)+ yg(x) = 1 with f and g being di.erentiable. 2
2. (15 Marks) Find the tangent line equation to the curve y = x +,
x
x> 0, that is perpendicular to the slant asymptote of the curve.
3. (15 Marks) Use the .-N language to prove lim n 1/n = 1. What is
n+
the limit lim (1 + n)1/n?
n+
4. (15 Marks) Show that the sequence 234 n
an =1+ + + + +
1!2!3! (n . 1)!
is convergent.
5. (15 Marks) If
an . an.1 = sin(/n),n =1, 2, 3,...,
does the sequence {an} converge? Please provide your justi.cation. 1. (40 Marks) Find the following quantities:
x . 1
n
(a) lim
x1
.
2 . 1
x
Solution
x . 1(
x . 1)(
nn
+ 1)
xn.1 +
xn.2 +
nn
lim
lim
=
2 . 1
(x . 1)(x + 1)(
n
xn.1 +
n
+ 1)
xn.2 +
x1
x1
x
1
= lim
x1
(x + 1)(
n
xn.1 +
n
+ 1)
xn.2 +
1
=
2n
1 . sin x 2
(b) lim .
x (x . )2
Solution
1 . sin x sin . sin x
2 22
lim = lim
x (x . )2 x (x . )2
2 sin(.x ) cos(+x )
= lim 44
x (x . )2
sin(.x ) 1 cos(+x ) . cos
4 42
= lim 2 (.1)
x (.x ) 4(x . )
4 (.2) sin(x. ) sin(x+3 )
1 88
=2 1 (.1) lim4 x (x . ) 1 sin(x. ) sin(x+3 )
88
=2 1 (.1) lim (.2)
4 x(x. )8
8
1 sin 1
2
=2 1 (.1) (.2) 1 = .
4 88
(c) lim [1+ xP (x)]1/x , where P (x) is a polynomial. Solution
x0
lim [1+ xP (x)]1/x = lim [1 + xP (x)]1/x
x0 x0
1
ln(1+xP (x))
= lim e
x0
x
1 xP (x)
P (x) ln(1+xP (x))
= lim e
x0
2
P (0) ln eP (0)
= e = e.
arcsin x
(d) lim . Solution
x0 sin x arcsin xy
lim = lim
x0 sin x y0 sin(sin y) y sin y
= lim =1 1=1.
y0 sin y sin(sin y)
(e) y., where xf(y)+ yg(x) = 1 with f and g being di.erentiable. Solution Di.erentiating both sides of the equation gives
f(y)+ xf.(y)y . + yg(x)+ yg .(x)=0.
Solving y. yields
f(y)+ yg.(x)
y . = . .
xf.(y)+ g(x)
2
2. (15 Marks) Find the tangent line equation to the curve y = x +,
x
x> 0, that is perpendicular to the slant asymptote of the curve. Solution Since
yx + 2
x
lim =lim =1,
x+ x x+ x
we know that the slope of the slant asymptote is 1. By the condition, the slope of the tangent line equation gives the equation
2
x += .1,
x
or
2
1 . = .1.
2
xSolving this equation gives x = 1 (neglecting x = .1 since x> 0). Thus the point on the curve is (1, 3). Thus, the tangent line equation is y =(.1)(x . 1) + 3,
or x + y = 4.
3. (15 Marks) Use the .-N language to prove lim n 1/n = 1. What is
n+
the limit lim (1 +