Q. Calculate the limit: lim (x -> 0) (x^3)/(sin(x)) (2023)
-
A.
0
-
B.
1
-
C.
∞
-
D.
Undefined
Solution
Using the fact that sin(x) ~ x as x approaches 0, we find that lim (x -> 0) (x^3)/(sin(x)) = 0.
Correct Answer: A — 0
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Q. Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4) (2023)
Solution
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer: A — 3/5
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Q. What is the limit: lim (x -> 0) (cos(x) - 1)/x^2? (2019)
-
A.
0
-
B.
-1/2
-
C.
1
-
D.
Undefined
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. What is the limit: lim (x -> 0) (e^x - 1)/x? (2022)
-
A.
1
-
B.
0
-
C.
e
-
D.
Undefined
Solution
Using the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = 1.
Correct Answer: A — 1
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