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Limits and continuity
NET · Mathematics · 5 questions
Name: ____________________ Date: ____________
What is the limit of (e^x - 1) / x as x tends to 0?
- A Infinity
- B e
- C 0
- D 1
What is the limit of (1 + 1/x)^x as x tends to infinity?
- A 1
- B e
- C 0
- D Infinity
What is the limit of (1 - cos x) / x² as x tends to 0?
- A 1/2
- B 2
- C 0
- D 1
The function f(x) = |x| at x = 0 is:
- A Neither continuous nor differentiable
- B Continuous but not differentiable
- C Differentiable but not continuous
- D Both continuous and differentiable
What is the limit of (x² - 9) / (x - 3) as x tends to 3?
- A 3
- B 6
- C 9
- D 0
Answer key
1. D, 2. B, 3. A, 4. B, 5. B
This is a standard limit and equals 1, since the derivative of e^x at 0 is 1.
This is the standard definition of the number e, about 2.718.
Using 1 - cos x = 2 sin²(x/2), the limit becomes 2 x (1/4) = 1/2.
The graph has no break at 0, so it is continuous, but it has a sharp corner there, so the derivative does not exist.
Factorising gives (x - 3)(x + 3) / (x - 3) = x + 3, which tends to 6.