Q. The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.
Correct Answer: B — No
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Q. The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:
-
A.
Continuous at x = 0
-
B.
Not continuous at x = 0
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C.
Continuous everywhere
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D.
None of the above
Solution
The function is not continuous at x = 0 since the limit does not equal f(0).
Correct Answer: B — Not continuous at x = 0
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Q. The function f(x) = { 2x + 3, x < 1; x^2 + 1, x >= 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the left limit (5) and the right limit (2). They are not equal, hence f(x) is not continuous at x = 1.
Correct Answer: B — 1
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Q. The function f(x) = { 3x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1 if which condition holds?
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A.
3 = 2
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B.
1 = 2
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C.
2 = 1
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D.
2 = 4
Solution
For continuity at x = 1, the left limit (3) must equal f(1) (2), which is not true.
Correct Answer: A — 3 = 2
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Q. The function f(x) = { 3x + 1, x < 1; 2x + 3, x >= 1 } is continuous at x = 1 if:
Solution
For continuity at x = 1, both pieces must equal 4, hence the function is continuous.
Correct Answer: A — 3
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Q. The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the left limit (3) and the right limit (3). Both equal 3, hence f(x) is continuous at x = 1.
Correct Answer: B — 1
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Q. The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.
Correct Answer: A — Yes
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Q. The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?
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A.
x = -1
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B.
x = 0
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C.
x = 1
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D.
x = 2
Solution
To check continuity at x = 0, we find f(0) = 1 and limit as x approaches 0 is also 1.
Correct Answer: B — x = 0
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Q. The function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } is:
-
A.
Continuous everywhere
-
B.
Continuous at x = 1
-
C.
Not continuous at x = 1
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D.
Continuous for x < 1
Solution
The function is not continuous at x = 1 because the left-hand limit does not equal the function value.
Correct Answer: C — Not continuous at x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at which point?
-
A.
x = 0
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B.
x = 1
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C.
x = 2
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D.
x = -1
Solution
To check continuity at x = 1, we find f(1) = 1, limit as x approaches 1 from left is 1, and from right is also 1.
Correct Answer: B — x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at x = ?
Solution
To check continuity at x = 1, we find the limit from both sides. Both limits equal 1, hence f(x) is continuous at x = 1.
Correct Answer: B — 1
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Q. The function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2 if:
-
A.
f(2) = 4
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B.
lim x->2 f(x) = 4
-
C.
Both a and b
-
D.
None of the above
Solution
Both conditions must hold true for continuity at x = 2.
Correct Answer: C — Both a and b
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Q. The function f(x) = { x^2, x < 2; k, x = 2; 3x - 4, x > 2 } is continuous at x = 2 for which value of k?
Solution
To be continuous at x = 2, k must equal f(2) = 2^2 = 4.
Correct Answer: C — 4
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Q. What is the limit of f(x) = 1/x as x approaches 0 from the right?
-
A.
0
-
B.
Infinity
-
C.
1
-
D.
Does not exist
Solution
As x approaches 0 from the right, f(x) approaches infinity, indicating a discontinuity at x = 0.
Correct Answer: B — Infinity
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Q. What is the value of k for which the function f(x) = { kx + 2, x < 2; x^2 - 4, x >= 2 is continuous at x = 2?
Solution
Setting 2k + 2 = 0 gives k = 2.
Correct Answer: C — 2
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Q. What is the value of k for which the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0?
Solution
Setting k(0) = 0^2 + 1 gives k = 1.
Correct Answer: B — 0
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Q. What is the value of k for which the function f(x) = { kx, x < 2; x^2, x >= 2 } is continuous at x = 2?
Solution
Setting k(2) = 2^2 gives 2k = 4, thus k = 2.
Correct Answer: C — 4
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Q. What is the value of p for which the function f(x) = { 3x + p, x < 2; x^2 - 4, x >= 2 } is continuous at x = 2?
Solution
Setting the two pieces equal at x = 2: 3(2) + p = 2^2 - 4. Solving gives p = -2.
Correct Answer: A — -1
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Q. What is the value of q for which the function f(x) = { 5 - q, x < 1; 3x + 2, x >= 1 } is continuous at x = 1?
Solution
Setting the two pieces equal at x = 1: 5 - q = 3(1) + 2. Solving gives q = 0.
Correct Answer: C — 2
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Q. What value of a makes the function f(x) = { 2x + 1, x < 1; a, x = 1; x^2 + 1, x > 1 continuous at x = 1?
Solution
Setting 2(1) + 1 = a and a = 2 for continuity.
Correct Answer: B — 2
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Q. What value of a makes the function f(x) = { 2x + a, x < 3; 5, x = 3; x^2 - 1, x > 3 continuous at x = 3?
Solution
Setting 2(3) + a = 5 gives a = -1.
Correct Answer: C — 2
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Q. What value of a makes the function f(x) = { 4 - x^2, x < 0; ax + 2, x = 0; x + 1, x > 0 continuous at x = 0?
Solution
Setting 4 = 2 gives a = 1 for continuity.
Correct Answer: B — 0
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Q. What value of k makes the function f(x) = { kx, x < 1; 2, x = 1; x + 1, x > 1 continuous at x = 1?
Solution
Setting the left limit (k(1) = k) equal to the right limit (1 + 1 = 2), we find k = 2.
Correct Answer: B — 1
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Q. What value of m makes the function f(x) = { 3x + 1, x < 2; mx + 4, x = 2; x^2 - 1, x > 2 continuous at x = 2?
Solution
Setting the left limit (3(2) + 1 = 7) equal to the right limit (2^2 - 1 = 3), we find m = 3.
Correct Answer: D — 4
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Q. Which of the following functions is continuous at x = 2?
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A.
f(x) = 1/x
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B.
f(x) = x^2 - 4
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C.
f(x) = sin(1/x)
-
D.
f(x) =
-
.
x
-
.
Solution
f(x) = x^2 - 4 is a polynomial function and is continuous everywhere, including at x = 2.
Correct Answer: B — f(x) = x^2 - 4
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Q. Which of the following functions is continuous at x = 2? f(x) = { x^2 - 4, x < 2; 3x - 6, x >= 2 }
-
A.
Continuous
-
B.
Not continuous
-
C.
Depends on k
-
D.
None of the above
Solution
At x = 2, f(2) = 0 and limit from left is 0, limit from right is also 0. Hence, it is continuous.
Correct Answer: A — Continuous
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Q. Which of the following functions is continuous at x = 2? f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }
-
A.
Continuous
-
B.
Not continuous
-
C.
Depends on k
-
D.
None of the above
Solution
To check continuity at x = 2, we find the left limit (4), right limit (4), and f(2) (4). All are equal, so f(x) is continuous.
Correct Answer: A — Continuous
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Q. Which of the following functions is continuous everywhere?
-
A.
f(x) = 1/x
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B.
f(x) = x^2
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C.
f(x) = sin(x)
-
D.
f(x) =
-
.
x
-
.
Solution
f(x) = x^2 is a polynomial function and is continuous everywhere.
Correct Answer: B — f(x) = x^2
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Q. Which of the following functions is not continuous at x = 0?
-
A.
f(x) = x^3
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B.
f(x) = e^x
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C.
f(x) = 1/x
-
D.
f(x) = ln(x)
Solution
The function f(x) = 1/x is not defined at x = 0, hence it is not continuous there.
Correct Answer: C — f(x) = 1/x
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Q. Which of the following functions is not continuous at x = 1?
-
A.
f(x) = x^2
-
B.
f(x) = 1/x
-
C.
f(x) = sin(x)
-
D.
f(x) = { x, x < 1; 2, x >= 1 }
Solution
The function has a jump discontinuity at x = 1, hence it is not continuous.
Correct Answer: D — f(x) = { x, x < 1; 2, x >= 1 }
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