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(MATH023)[2006](f)midterm~dli^_10402.pdf
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Math 023 -Advanced Mathematics I Midterm Test, Fall Semester, 2006
Time Allowed: 2 Hours Total Marks: 100
1. (40 Marks) Evaluate the following limits:
x
(a) lim ;
x0 1+ x . 1
2
(b) lim (1 + 1/x)x+11/x;
x+1
sin x + sin 2x + + sin nx
(c) lim ;
x0 n2x
sin x + sin 2x + + sin nx

(d) lim .
n1 n2x
2. (20 Marks) For what values of a, b and c, does the function y =

ax2 + bx + c have an asymptote when x approaches +1? Please .nd the equation of the asymptote.
3. (10 Marks) Let
f(x)=

...
..
sin x
, if x =6;
x .
A, if x = .
This function is de.ned everywhere on the real axis. Determine the value A so that f is continuous everywhere. Use - language to justify your argument.
4. (30 Marks)
(a) For any x0 (.1, +1), show that the sequence {xn} generated by
1
xn = + sin xn.1,n =1, 2,...,
2
is a Cauchy sequence.
(b) Show that the above sequence {xn} converges to the unique so-
lution of the equation
1

x = + sin x.
2
(c) Show that the unique solution is positive.
1