Department

MathematicsCourse Code

MAT137Y1Professor

N/ AStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**2 pages of the document.**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.

1

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