Limits and Continuity Quiz

Grade 12 · Math · Quiz

10 questions covering limit evaluation, continuity conditions, and discontinuity types.

Questions

Evaluate lim(x→3) (x² - 9)/(x - 3).
1. 0
2. 3
3. 6
4. 9
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Answer: 6
(x²-9)/(x-3) = (x+3)(x-3)/(x-3) = x+3. At x=3: 3+3 = 6.
A function f has lim(x→1⁻) f(x) = 2 and lim(x→1⁺) f(x) = 5. What type of discontinuity is at x = 1?
1. Removable
2. Jump
3. Infinite
4. None
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Answer: Jump
The left and right limits both exist but are unequal (2 ≠ 5), indicating a jump discontinuity.
What is lim(x→∞) (5x³ - 2x)/(2x³ + x²)?
1. 0
2. 5/2
3. 5
4. ∞
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Answer: 5/2
Dominant terms: 5x³/2x³ = 5/2. Lower-degree terms vanish as x→∞.
Every continuous function is also differentiable.
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Continuity does not imply differentiability. For example, f(x) = |x| is continuous at x=0 but not differentiable there.
lim(x→0) sin(x)/x equals:
1. 0
2. 1
3. ∞
4. Undefined
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Answer: 1
This is a standard limit: lim(x→0) sin(x)/x = 1.
For f(x) = (x² - 1)/(x - 1), what value of f(1) makes f continuous at x = 1?
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lim(x→1) (x²-1)/(x-1) = lim(x→1)(x+1) = 2. Setting f(1) = 2 fills the hole.
Which of the following functions is continuous everywhere on ℝ?
1. f(x) = 1/x
2. f(x) = √x
3. f(x) = sin(x)
4. f(x) = tan(x)
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Answer: f(x) = sin(x)
sin(x) is defined and continuous for all real numbers. The others have domain restrictions or discontinuities.
lim(x→a) f(x) = f(a) is sufficient to guarantee continuity of f at x = a.
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This single equality encapsulates all three continuity conditions: f(a) exists, the limit exists, and they are equal.
What is lim(x→2) (x³ - 8)/(x - 2)?
1. 4
2. 8
3. 12
4. 0
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Answer: 12
x³-8 = (x-2)(x²+2x+4). Cancel (x-2): lim = x²+2x+4 at x=2 = 4+4+4 = 12.
Evaluate lim(x→∞) (2x + 3)/(x² + 1).
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The denominator grows faster (degree 2 vs degree 1), so the limit is 0.

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