ECAT Mathematics
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ECAT Mathematics
Chapter 15: Functions and Limits
84 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
lim_{x→3} (x²−9)/(x−3) =
A
0
B
3
C
6
D
9
1
lim_{x→0} sinx/x =
A
0
B
∞
C
1
D
x
2
lim_{x→∞} 1/x =
A
1
B
∞
C
0
D
−1
3
lim_{x→0} (1−cosx)/x =
A
0
B
1
C
1/2
D
∞
4
lim_{x→0} (1−cosx)/x² =
A
0
B
1
C
1/2
D
∞
5
lim_{x→0} tanx/x =
A
0
B
1/2
C
1
D
∞
6
lim_{x→∞} (3x²+2x)/(x²+1) =
A
2
B
3
C
∞
D
0
7
lim_{x→2} (x²−4)/(x−2) =
A
0
B
2
C
4
D
8
8
A function f is continuous at x=a if:
A
f(a) is undefined
B
lim_{x→a}f(x) = f(a)
C
f(a) ≠ lim
D
f is differentiable
9
f(x)=1/x is discontinuous at:
A
x=1
B
x=0
C
x=−1
D
x=∞
10
lim_{x→0⁺} 1/x =
A
0
B
−∞
C
∞
D
1
11
lim_{x→0⁻} 1/x =
A
0
B
+∞
C
−∞
D
1
12
lim_{x→0} (e^x−1)/x =
A
0
B
e
C
1
D
∞
13
lim_{x→∞} (1+1/x)^x =
A
1
B
e
C
∞
D
0
14
lim_{x→a} [f(x)±g(x)] =
A
lim f(x) × lim g(x)
B
lim f(x) ± lim g(x)
C
lim f(x)/lim g(x)
D
undefined
15
lim_{x→a} [f(x)g(x)] =
A
lim f(x) + lim g(x)
B
lim f(x) × lim g(x)
C
lim f(x)/lim g(x)
D
undefined
16
lim_{x→a} [f(x)/g(x)] = lim f(x)/lim g(x) provided:
A
lim g(x)=0
B
lim g(x)≠0
C
lim f(x)=0
D
Always valid
17
L'Hopital's rule applies when:
A
limit exists
B
limit gives 0/0 or ∞/∞
C
limit gives 0×∞
D
Any indeterminate form
18
lim_{x→0} x/sinx =
A
0
B
1
C
∞
D
1/2
19
f(x) = x² is continuous:
A
Only at x=0
B
Everywhere
C
Only for x>0
D
Nowhere
20
A polynomial is continuous:
A
Nowhere
B
Only at integer points
C
Everywhere (on its domain=all reals)
D
Only for positive x
21
A rational function p(x)/q(x) is discontinuous where:
A
p(x)=0
B
q(x)=0
C
p(x)=q(x)
D
Both=0
22
lim_{x→1} (x³−1)/(x−1) =
A
0
B
1
C
2
D
3
23
lim_{x→4} √x =
A
2
B
√2
C
4
D
∞
24
The type of function f(x) = ⌊x⌋ (floor) is:
A
Continuous everywhere
B
Discontinuous at all integers
C
Differentiable everywhere
D
Only defined for integers
25
Sandwich (Squeeze) theorem: if g(x)≤f(x)≤h(x) and lim g=lim h=L, then lim f=
A
0
B
L
C
g(x)
D
∞
26
lim_{x→0} x sin(1/x) = ?
A
1
B
0
C
∞
D
−1
27
lim_{x→0} sin(3x)/x =
A
1
B
3
C
1/3
D
0
28
lim_{x→0} sin(ax)/(bx) =
A
a/b
B
b/a
C
1
D
ab
29
lim_{x→∞} sinx/x =
A
1
B
0
C
∞
D
−1
30
A removable discontinuity can be fixed by:
A
Nothing
B
Redefining f at that point
C
Changing domain
D
Multiplying by a constant
31
lim_{x→0} (tan3x)/(tan5x) =
A
3/5
B
5/3
C
1
D
15
32
A jump discontinuity occurs when:
A
Left and right limits exist but are unequal
B
Left limit does not exist
C
f is undefined
D
f is continuous
33
An infinite discontinuity (like 1/x at 0) has:
A
Equal one-sided limits
B
One or both one-sided limits = ±∞
C
Both limits exist and equal
D
No discontinuity
34
f(x)={x+1 if x<2; 3 if x=2; x+1 if x>2} is:
A
Discontinuous at x=2
B
Continuous at x=2
C
Not a function
D
Undefined
35
For f(x) to be continuous at x=a, we need: f(a) exists, lim exists, and:
A
lim = 0
B
lim = f(a)
C
lim = a
D
lim = ∞
36
Intermediate Value Theorem: if f is continuous on [a,b] and f(a)<k<f(b), then there exists c in (a,b) with:
A
f(c)=a
B
f(c)=b
C
f(c)=k
D
f(c)=0
37
lim_{x→−∞} (2x+3)/(4x−1) =
A
0
B
1/2
C
2
D
∞
38
lim_{x→∞} (x²+x)/(2x²−3) =
A
0
B
1/2
C
1
D
2
39
lim_{x→0} (√(1+x)−1)/x =
A
0
B
1/2
C
1
D
∞
40
lim_{x→π} (sinx)/(x−π) = ? (let u=x−π: sin(u+π)/u=−sinu/u→−1)
A
1
B
−1
C
0
D
π
41
The limit lim_{x→0} (a^x−1)/x = lna. For a=e:
A
lne=1
B
lne=0
C
lne=e
D
lne=2
42
The limit of a constant function f(x)=k as x→a:
A
0
B
k
C
a
D
undefined
43
lim_{x→0} x²/|x| =
A
0
B
1
C
−1
D
undefined
44
f(x)={x² if x≠0; 1 if x=0}. f is discontinuous at x=0 because:
A
f(0) undefined
B
lim=0≠f(0)=1
C
f is not defined
D
lim does not exist
45
lim_{x→0⁺} lnx =
A
0
B
−∞
C
∞
D
1
46
lim_{x→∞} lnx =
A
0
B
−∞
C
∞
D
1
47
lim_{x→0} (2^x−1)/x =
A
0
B
1
C
ln2
D
2
48
Right-hand limit lim_{x→a⁺}f(x): x approaches a from:
A
Left
B
Right
C
Both sides simultaneously
D
Neither
49
A function has a limit at x=a iff:
A
Only f(a) exists
B
Left-hand limit = right-hand limit
C
f(a)=0
D
f is differentiable
50
lim_{x→∞} e^x =
A
0
B
1
C
∞
D
e
51
lim_{x→−∞} e^x =
A
0
B
1
C
∞
D
e
52
lim_{x→0} (cosx−1)/x² = ?
A
−1/2
B
1/2
C
0
D
1
53
A function f is even if:
A
f(−x)=−f(x)
B
f(−x)=f(x)
C
f(x)=0
D
f is constant
54
A function f is odd if:
A
f(−x)=f(x)
B
f(−x)=−f(x)
C
f(0)=0
D
Both B and C might hold
55
Domain of f(x)=√(4−x²):
A
−2≤x≤2
B
x>2
C
x<−2
D
All reals
56
Range of f(x)=x²+1:
A
All reals
B
[0,∞)
C
[1,∞)
D
(1,∞)
57
lim_{x→0} sin²x/x² =
A
0
B
1
C
1/2
D
∞
58
A vertical asymptote of f(x)=1/(x−2) is at:
A
x=2
B
y=2
C
x=0
D
y=0
59
A horizontal asymptote of f(x)=(3x+1)/(x+2) as x→∞:
A
y=1
B
y=2
C
y=3
D
y=0
60
An oblique asymptote occurs when:
A
degree(num)=degree(den)
B
degree(num)>degree(den) by 1
C
degree(num)<degree(den)
D
degree(num)=0
61
lim_{x→3} (x²−9)/(x²−3x) = lim_{x→3} (x+3)/(x):
A
2
B
3
C
6
D
∞
62
lim_{x→0} (3x+sinx)/x = lim(3+sinx/x)→
A
3+1=4
B
3
C
1
D
4
63
The Extreme Value Theorem: f continuous on [a,b] attains:
A
Only minimum
B
Only maximum
C
Both maximum and minimum
D
Neither necessarily
64
f(x)=|x| is:
A
Not continuous
B
Continuous everywhere
C
Differentiable everywhere
D
Discontinuous at x=0
65
lim_{x→0} |x|/x =
A
1
B
0
C
Does not exist
D
−1
66
A function is bounded if:
A
It has a maximum only
B
|f(x)|≤M for some M and all x in domain
C
It is continuous
D
It is differentiable
67
f(x)=x³ is:
A
Even
B
Odd
C
Neither
D
Constant
68
The composition f(g(x)) exists if:
A
Range of g ⊆ domain of f
B
Domain of g = domain of f
C
f and g are equal
D
f is bijective
69
lim_{x→1} (x^n−1)/(x−1) =
A
n−1
B
n
C
0
D
1
70
lim_{x→∞} x^{1/x} =
A
0
B
1
C
∞
D
e
71
A function with a hole (removable discontinuity) at x=a has:
A
No limit at x=a
B
A limit at x=a equal to f(a)
C
A limit at x=a ≠ f(a) or f(a) undefined
D
A vertical asymptote
72
lim_{x→0} x^x =
A
0
B
1
C
∞
D
undefined
73
f(x) = c (constant): lim_{x→a} f(x) =
A
a
B
0
C
c
D
∞
74
lim_{h→0} [f(x+h)−f(x)]/h is the definition of:
A
Integral
B
Derivative
C
Limit at infinity
D
Continuity
75
For f(x)=2x: lim_{h→0} [2(x+h)−2x]/h =
A
2h/h=2
B
2x
C
0
D
∞
76
The limit lim_{x→a}f(x) may or may not equal f(a). It equals f(a) when:
A
f is differentiable only
B
f is continuous at a
C
f is bounded
D
f is odd
77
lim_{x→0} (e^{2x}−1)/(x) = ?
A
1
B
2
C
e
D
0
78
lim_{x→0} ln(1+x)/x = ? (standard limit)
A
0
B
e
C
1
D
∞
79
lim_{x→0⁺} x^x (same as above) =
A
0
B
1
C
∞
D
undefined
80
If lim_{x→a}f(x)=L and lim_{x→a}g(x)=M, then lim f(x)×g(x) =
A
L+M
B
LM
C
L/M
D
L−M
81
The formal ε−δ definition states: lim_{x→a}f(x)=L if for every ε>0, there exists δ>0 such that:
A
|f(x)−a|<ε
B
|x−a|<δ implies |f(x)−L|<ε
C
|f(x)|<L
D
x<a+δ
82
lim_{x→1} (x^2−1)/(x−1) =
A
0
B
1
C
2
D
4
83
lim_{x→0} [sin(5x)/sin(3x)] = ?
A
5/3
B
3/5
C
15
D
1/15
0
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