Q. Calculate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
-
A.
0
-
B.
1/2
-
C.
1
-
D.
Infinity
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1.
Correct Answer: B — 1/2
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Q. Calculate the limit: lim (x -> 0) (e^x - 1)/x
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Using the definition of the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = e^0 = 1.
Correct Answer: B — 1
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Q. Calculate the limit: lim (x -> 0) (tan(3x)/x)
-
A.
3
-
B.
1
-
C.
0
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 3, so the limit is 3.
Correct Answer: A — 3
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.
Correct Answer: C — 2
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)^2
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
Factoring gives (x - 1)(x + 1)/(x - 1)^2 = (x + 1)/(x - 1). Thus, lim (x -> 1) (x + 1)/(x - 1) = 2.
Correct Answer: C — 2
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Q. Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
-
A.
0
-
B.
1
-
C.
3
-
D.
Undefined
Solution
Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim (x -> 1) (x^2 + x + 1) = 3.
Correct Answer: C — 3
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Q. Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)
-
A.
0
-
B.
2
-
C.
4
-
D.
Undefined
Solution
Factoring gives (x(x - 2))/(x - 2), canceling gives lim (x -> 2) x = 2.
Correct Answer: D — Undefined
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Q. Evaluate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
-
A.
0
-
B.
1/2
-
C.
1
-
D.
Undefined
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1/2.
Correct Answer: B — 1/2
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Q. Evaluate the limit: lim (x -> 0) (e^x - 1)/x
-
A.
0
-
B.
1
-
C.
e
-
D.
Infinity
Solution
Using L'Hôpital's Rule, we differentiate the numerator and denominator: lim (x -> 0) (e^x)/(1) = e^0 = 1.
Correct Answer: B — 1
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Q. Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Using L'Hôpital's Rule, differentiate the numerator and denominator: lim (x -> 0) (1/(1 + x))/(1) = 1.
Correct Answer: B — 1
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Q. Evaluate the limit: lim (x -> 0) (sin(5x)/x)
-
A.
0
-
B.
5
-
C.
1
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (sin(x)/x) = 1, we have lim (x -> 0) (sin(5x)/x) = 5 * lim (x -> 0) (sin(5x)/(5x)) = 5 * 1 = 5.
Correct Answer: B — 5
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Q. Evaluate the limit: lim (x -> 0) (tan(3x)/x)
-
A.
0
-
B.
3
-
C.
1
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (tan(x)/x) = 1, we have lim (x -> 0) (tan(3x)/x) = 3 * lim (x -> 0) (tan(3x)/(3x)) = 3 * 1 = 3.
Correct Answer: B — 3
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Q. Evaluate the limit: lim (x -> 0) (x^2 * sin(1/x))
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Since |sin(1/x)| <= 1, we have |x^2 * sin(1/x)| <= x^2, and thus lim (x -> 0) x^2 * sin(1/x) = 0.
Correct Answer: A — 0
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Q. Evaluate the limit: lim (x -> 0) (x^2)/(sin(x))
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
As x approaches 0, sin(x) approaches x, thus lim (x -> 0) (x^2)/(sin(x)) = 0.
Correct Answer: A — 0
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Q. Evaluate the limit: lim (x -> ∞) (2x^3 - 3x)/(4x^3 + 5)
Solution
Dividing numerator and denominator by x^3 gives lim (x -> ∞) (2 - 3/x^2)/(4 + 5/x^3) = 2/4 = 1/2.
Correct Answer: B — 1/2
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Q. Evaluate the limit: lim(x->1) (x^2 - 1)/(x - 1)^2
-
A.
1
-
B.
2
-
C.
0
-
D.
Undefined
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1)^2 = (x+1)/(x-1). Thus, lim(x->1) = 2.
Correct Answer: B — 2
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Q. Evaluate the limit: lim(x->infinity) (2x^3 - 3x)/(4x^3 + 5)
-
A.
1/2
-
B.
0
-
C.
1
-
D.
Infinity
Solution
Divide numerator and denominator by x^3: lim(x->infinity) (2 - 3/x^2)/(4 + 5/x^3) = 2/4 = 1/2.
Correct Answer: A — 1/2
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Q. Evaluate the limit: lim(x->infinity) (3x^2 + 2)/(5x^2 - 4)
-
A.
3/5
-
B.
0
-
C.
1
-
D.
Infinity
Solution
Divide numerator and denominator by x^2: lim(x->infinity) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer: A — 3/5
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Q. Find the limit: lim (x -> 0) (1 - cos(2x))/x^2
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
Using the identity 1 - cos(θ) = 2sin^2(θ/2), we have lim (x -> 0) (1 - cos(2x))/x^2 = lim (x -> 0) (2sin^2(x))/x^2 = 2.
Correct Answer: C — 2
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Q. Find the limit: lim (x -> 0) (1 - cos(4x))/(x^2)
-
A.
0
-
B.
2
-
C.
4
-
D.
Undefined
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(2x))/(x^2) = 8.
Correct Answer: B — 2
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Q. Find the limit: lim (x -> 0) (1 - cos(x))/(x^2)
-
A.
0
-
B.
1/2
-
C.
1
-
D.
Infinity
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1.
Correct Answer: B — 1/2
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Q. Find the limit: lim (x -> 0) (cos(x) - 1)/x^2
-
A.
0
-
B.
-1/2
-
C.
1
-
D.
Infinity
Solution
Using the Taylor series expansion for cos(x), we find that lim (x -> 0) (cos(x) - 1)/x^2 = -1/2.
Correct Answer: B — -1/2
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Q. Find the limit: lim (x -> 0) (e^x - 1)/x
-
A.
0
-
B.
1
-
C.
e
-
D.
Undefined
Solution
Using the derivative of e^x at x = 0, we find lim (x -> 0) (e^x - 1)/x = 1.
Correct Answer: B — 1
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Q. Find the limit: lim (x -> 0) (x^2 * sin(1/x))
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Since |sin(1/x)| <= 1, we have |x^2 * sin(1/x)| <= |x^2|. As x approaches 0, |x^2| approaches 0, hence the limit is 0.
Correct Answer: A — 0
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Q. Find the limit: lim (x -> 0) (x^3)/(e^x - 1)
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
As x approaches 0, e^x - 1 approaches 0. Using L'Hôpital's Rule three times, we find the limit approaches 0.
Correct Answer: A — 0
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Q. Find the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.
Correct Answer: C — 2
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Q. Find the limit: lim (x -> 1) (x^2 - 1)/(x - 1)^2
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
Factoring gives (x - 1)(x + 1)/(x - 1)^2. Canceling gives lim (x -> 1) (x + 1)/(x - 1) = 2.
Correct Answer: C — 2
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Q. Find the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
Solution
Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^2 + x + 1) = 3.
Correct Answer: C — 3
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Q. Find the limit: lim (x -> 2) (x^2 - 4)/(x - 2)
-
A.
0
-
B.
2
-
C.
4
-
D.
Undefined
Solution
The expression (x^2 - 4)/(x - 2) can be factored as (x - 2)(x + 2)/(x - 2). Canceling (x - 2) gives lim (x -> 2) (x + 2) = 4.
Correct Answer: C — 4
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Q. Find the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4)
-
A.
0
-
B.
3/5
-
C.
1
-
D.
Infinity
Solution
Dividing numerator and denominator by x^2, we get lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer: B — 3/5
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