Department of Mathematics, University of Toronto
Term Test 1–November 15, 2006
MAT137Y,Calculus!
Time Alloted: 1hour 50 minutes
1. Evaluate the following limits. Do not use L’H ˆ
opital’sRule to evaluate the limit.
(7%) (i) lim
x→0
(x−1)2−1
x2+6x
.
(7%) (ii) lim
t→0
sin2(5t)
3t2.
(7%) (iii) lim
x→4+
(4−x)|3x−14|
|4−x|
.
(7%) (iii) lim
x→0
3−
√
9−x2
x2.
2.
(7%) (i) Solvethe inequality
x2−3x
x4−1
≤0. Express your answer as aunion of intervals.
(ii) Suppose sin x=3
4and π
2≤x≤π.Find the exact value of each of the following expres-
sions.
(6%) (a) tan x.
(4%) (b) cos 2x.
3.
(5%) (a) Givethe precise ε,δdefinition of the following statement: lim
x→a
f(x)=L.
(12%) (b) Provethat lim
x→3
x2+1
1−x
=−5directly using the precise definition of limit.
4. Consider the sequence of numbers
x1=
√
1,x2=
q
1+
√
1,x3=
r
1+
q
1+
√
1,x4=
s
1+
r
1+
q
1+
√
1,...,
so xncontaines nnested radicals and exactly nones.
(3%) (a) Express xnin terms of xn−1.
(9%) (b) Provefor all positiveintegers n≥2that xnis irrational.
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