Continuity in Education for Competitive Exam Success

Continuity

Q. For which value of b is the function f(x) = { 2x + 1, x < 1; b, x = 1; x^2 + 1, x > 1 continuous at x = 1?

A. 1
B. 2
C. 3
D. 4

Solution

Setting the left limit (2(1) + 1 = 3) equal to the right limit (1^2 + 1 = 2), we find b = 3.

Correct Answer: B — 2

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Q. For which value of b is the function f(x) = { x^2 - 1, x < 1; b, x = 1; 3x - 2, x > 1 continuous at x = 1?

A. 0
B. 1
C. 2
D. 3

Solution

Setting limit as x approaches 1 gives b = 2 for continuity.

Correct Answer: C — 2

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Q. For which value of b is the function f(x) = { x^2 - 4, x < 2; bx + 2, x >= 2 } continuous at x = 2?

A. 2
B. 4
C. 6
D. 8

Solution

Setting the two pieces equal at x = 2 gives us 0 = 2b + 2. Solving for b gives b = -1.

Correct Answer: B — 4

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Q. For which value of b is the function f(x) = { x^3 - 3x + b, x < 1; 2x + 1, x >= 1 continuous at x = 1?

A. 0
B. 1
C. 2
D. 3

Solution

Setting 1 - 3 + b = 2 gives b = 4 for continuity.

Correct Answer: A — 0

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Q. For which value of c is the function f(x) = { 3x + c, x < 1; 2x^2, x >= 1 continuous at x = 1?

A. -1
B. 0
C. 1
D. 2

Solution

Setting 3(1) + c = 2(1)^2 gives c = -1 for continuity.

Correct Answer: B — 0

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Q. For which value of c is the function f(x) = { x^2 - 4, x < c; 3x - 5, x >= c } continuous at x = c?

A. 1
B. 2
C. 3
D. 4

Solution

Setting the two pieces equal at x = c: c^2 - 4 = 3c - 5. Solving gives c = 3.

Correct Answer: C — 3

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Q. For which value of c is the function f(x) = { x^2 - c, x < 1; 2x + 1, x >= 1 continuous at x = 1?

A. 0
B. 1
C. 2
D. 3

Solution

Setting x^2 - c = 2x + 1 at x = 1 gives c = 2.

Correct Answer: C — 2

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Q. For which value of c is the function f(x) = { x^2, x < c; 2x + 1, x >= c continuous at x = c?

A. 0
B. 1
C. 2
D. 3

Solution

Setting x^2 = 2x + 1 at x = c gives c = 2.

Correct Answer: C — 2

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Q. For which value of m is the function f(x) = { 3x + m, x < 1; 2, x = 1; mx + 1, x > 1 continuous at x = 1?

A. -1
B. 0
C. 1
D. 2

Solution

Setting 3 + m = 2 and 2 = m + 1 gives m = 1 for continuity.

Correct Answer: B — 0

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Q. For which value of p is the function f(x) = { x^2 + p, x < 0; 3x - 1, x >= 0 } continuous at x = 0?

A. -1
B. 0
C. 1
D. 2

Solution

Setting the two pieces equal at x = 0 gives us p = -1.

Correct Answer: B — 0

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Q. If f(x) = 1/(x-1), what is the point of discontinuity?

A. x = 0
B. x = 1
C. x = -1
D. x = 2

Solution

The function is discontinuous at x = 1 because it leads to division by zero.

Correct Answer: B — x = 1

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Q. If f(x) = x^2 - 4, what is the limit of f(x) as x approaches 2?

A. 0
B. 2
C. 4
D. Undefined

Solution

The limit as x approaches 2 is f(2) = 2^2 - 4 = 0.

Correct Answer: C — 4

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Q. If f(x) = x^3 - 3x + 2, then f(x) is continuous at:

A. All x
B. x = 0
C. x = 1
D. x = -1

Solution

f(x) is a polynomial function and is continuous for all x.

Correct Answer: A — All x

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Q. If f(x) = { 2x + 3, x < 0; kx + 1, x >= 0 } is continuous at x = 0, what is the value of k?

A. -3/2
B. 1/2
C. 3/2
D. 2

Solution

Setting the two pieces equal at x = 0: 3 = k(0) + 1. Solving gives k = -3/2.

Correct Answer: A — -3/2

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Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 } is continuous at x = 0, what is k?

A. 1
B. 0
C. 2
D. 3

Solution

For continuity at x = 0, we need the left limit (1) to equal k. Thus, k = 1.

Correct Answer: A — 1

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Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 }, what value of k makes f continuous at x = 0?

A. 0
B. 1
C. 2
D. 3

Solution

To be continuous at x = 0, k must equal the limit from the left, which is 1.

Correct Answer: B — 1

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Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x, x > 0 }, for f(x) to be continuous at x = 0, k must be:

A. 0
B. 1
C. 2
D. 3

Solution

For continuity at x = 0, k must equal the limit as x approaches 0, which is 1.

Correct Answer: B — 1

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Q. If f(x) = { x^2 + 1, x < 0; kx + 2, x = 0; 3 - x, x > 0 is continuous at x = 0, find k.

A. 1
B. 2
C. 3
D. 4

Solution

For continuity at x = 0, we need 1 = 2, thus k must be 1.

Correct Answer: B — 2

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Q. If f(x) = { x^2 + 1, x < 0; kx + 3, x = 0; 2x - 1, x > 0 is continuous at x = 0, find k.

A. -1
B. 0
C. 1
D. 2

Solution

For continuity at x = 0, we need 1 = 3 and 1 = -1 + 3k, solving gives k = 1.

Correct Answer: C — 1

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Q. If f(x) = { x^2, x < 0; 2x + 3, x >= 0 }, find f(0).

A. 0
B. 3
C. 1
D. undefined

Solution

At x = 0, we use the second piece: f(0) = 2(0) + 3 = 3.

Correct Answer: B — 3

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Q. If f(x) = { x^2, x < 0; kx + 1, x = 0; 2x + 3, x > 0 is continuous at x = 0, find k.

A. -1
B. 0
C. 1
D. 2

Solution

To ensure continuity at x = 0, we set k(0) + 1 = 0^2, leading to k = 2.

Correct Answer: C — 1

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Q. If f(x) = { x^2, x < 1; kx + 1, x >= 1 } is continuous at x = 1, find k.

A. 0
B. 1
C. 2
D. 3

Solution

Setting the two pieces equal at x = 1 gives 1 = k + 1, hence k = 0.

Correct Answer: B — 1

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Q. If f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2, what is the value of f(2)?

A. 2
B. 4
C. 3
D. 5

Solution

For continuity at x = 2, f(2) must equal the limit from both sides, which is 4.

Correct Answer: B — 4

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Q. If f(x) = { x^2, x < 3; k, x = 3; 2x, x > 3 } is continuous at x = 3, what is the value of k?

A. 3
B. 9
C. 6
D. 0

Solution

For continuity at x = 3, we need limit as x approaches 3 from left (9) to equal f(3) = k, thus k = 9.

Correct Answer: C — 6

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Q. If f(x) = { x^2, x < 3; k, x = 3; 3x - 2, x > 3 } is continuous at x = 3, what is k?

A. 7
B. 9
C. 8
D. 6

Solution

For continuity at x = 3, we need k to equal the limit from both sides, which is 9.

Correct Answer: C — 8

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Q. If f(x) is continuous on [a, b], which of the following must be true?

A. f(a) = f(b)
B. f(x) takes every value between f(a) and f(b)
C. f(x) is increasing
D. f(x) is decreasing

Solution

By the Intermediate Value Theorem, a continuous function on a closed interval takes every value between f(a) and f(b).

Correct Answer: B — f(x) takes every value between f(a) and f(b)

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Q. Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x = 0?

A. Yes
B. No
C. Only left continuous
D. Only right continuous

Solution

Both limits as x approaches 0 from the left and right are equal to 1, hence f(x) is continuous at x = 0.

Correct Answer: A — Yes

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Q. Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?

A. Yes
B. No
C. Depends on x
D. Not defined

Solution

Both limits as x approaches 0 from the left and right are equal to 0, hence f(x) is continuous at x = 0.

Correct Answer: A — Yes

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Q. Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?

A. Yes
B. No
C. Only left continuous
D. Only right continuous

Solution

Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.

Correct Answer: A — Yes

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Q. Is the function f(x) = |x|/x continuous at x = 0?

A. Yes
B. No
C. Depends on direction
D. None of the above

Solution

The left limit is -1 and the right limit is 1, which are not equal. Therefore, f(x) is not continuous at x = 0.

Correct Answer: B — No

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Continuity

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