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Limits and continuity
5 NET Mathematics questions with answers and explanations.
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What is the limit of (e^x - 1) / x as x tends to 0?
- 1 0
- 2 1
- 3 e
- 4 Infinity
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Correct answer: 1
This is a standard limit and equals 1, since the derivative of e^x at 0 is 1.
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What is the limit of (1 + 1/x)^x as x tends to infinity?
- 1 1
- 2 0
- 3 e
- 4 Infinity
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Correct answer: e
This is the standard definition of the number e, about 2.718.
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What is the limit of (1 - cos x) / x^2 as x tends to 0?
- 1 0
- 2 1/2
- 3 1
- 4 2
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Correct answer: 1/2
Using 1 - cos x = 2 sin^2(x/2), the limit becomes 2 x (1/4) = 1/2.
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The function f(x) = |x| at x = 0 is:
- 1 Neither continuous nor differentiable
- 2 Continuous but not differentiable
- 3 Differentiable but not continuous
- 4 Both continuous and differentiable
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Correct answer: Continuous but not differentiable
The graph has no break at 0, so it is continuous, but it has a sharp corner there, so the derivative does not exist.
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What is the limit of (x^2 - 9) / (x - 3) as x tends to 3?
- 1 0
- 2 3
- 3 6
- 4 9
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Correct answer: 6
Factorising gives (x - 3)(x + 3) / (x - 3) = x + 3, which tends to 6.