MATH 136
Practice Test 1
Answers
1. (i) Evaluate the limit. (ii) Completely describe the limit in words in terms of how
the variable and the function decrease/increase.
lim 5cot x = 5
(a)
x→
π
4
(b)
+
As x decreases to π/4,
5cot x increases to 5.
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lim
x → −6−
3x
36 − x
2
=
−18
= +∞
−0
As x increases to –6,
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3x /(36 − x 2 ) increases to +∞.
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2. Evaluate the limits:
 3cos(π x) if x < 4

5
if x = 4
(i) Let f (x) = 
 7sin(π x /8) if x > 4 .

(a)
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lim f (x)
(b) lim f (x)
x → 4–
x → 4+
= 3cos(4π) = 3
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 x2

(ii) Let g(x) =  2 − x
 10

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if x < −2
if x > −2
if x = −2
lim g(x)
(b)
x → −2 –
2
lim g(x)
x → −2 +
(c)
= 2 − (−2) = 4
= (−2) = 4
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x→4
= 7sin(π /2) = 7
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(a)
(c) lim f (x) does not exist
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lim g(x) = 4
x → −2
3. State what type of indeterminate expression results from attempting to evaluate the
limit. Then find the limit with appropriate values of x . State the values of x used.
3tan x
x→π x −π
0
gives
0
(b)
(a) lim
f (π − 0.00001) = 3
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f (π + 0.00001) = 3
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f (−π /2 − 0.00001) = 0.999995
The limit appears to be 1.
(d) lim (sin(2x)) × cot x
x → 0−
x → 0+
gives 0 × (−∞)
gives 1∞
f (−0.0001) = 1.9999...
f (0.00001) = 1.000005
The limit appears to be 1.
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π_
2
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(c) lim (sec x) csc x
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x →−
(sin(−x)) tan(x)
gives 1∞
The limit appears to be 3.
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lim
Use Y1 = (1/cos(x)) (1/sin(x))
in radian mode
f (−0.00001) = 2
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The limit appears to be 2.
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€ = (sin(2x)) /tan(x)
Use Y1
in radian mode
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4. Evaluate the limits.
6 x −2
x −2
x → 2−
6 x −2
x −2
x → 2+
(a) lim
(c) lim
does not exist
=6
= −6
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6 x −2
x→2 x −2
(b) lim
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