You need to explain all reasons in detail and show all of your work on
the answer sheet; otherwise you will get NO credits. If you used any theorem
in textbook or proved in class, state it carefully and explicitly.
I.(20%) Evaluate lim f(x) if it exists, where
x→3
｜ x^2, x<3
f(x) = ｜ 7, x=3
｜2x+3, x>3
II.(20%) Given that f(x) = x^2-4x, evaluate the following limit if it exists.
f(x)-f(2)
lim ───────
x→3 x-3
III.(30%) Given necessary and sufficient conditions on A and B for the
function.
｜ Ax-B, x<=1
f(x) = ｜ 3x, 1<x<2
｜Bx^2-A, 2<=x
IV.(30%) Let f be the Dirichlet function
f(x) = ｜ 1, x rational
｜ 0, x irrational
Show that lim xf(x) = 0
x→0
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