Limits & Continuity
Q. Calculate the limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)
Solution
Dividing the numerator and denominator by x^2, we get lim (x -> ∞) (5 + 3/x^2)/(2 + 1/x^2) = 5/2.
Correct Answer: B — 5/2
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Q. Evaluate the limit: lim (x -> 0) (tan(x)/x) (2023)
-
A.
0
-
B.
1
-
C.
∞
-
D.
Undefined
Solution
Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.
Correct Answer: B — 1
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Q. Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)
-
A.
0
-
B.
1
-
C.
4
-
D.
Undefined
Solution
Factoring gives ((x - 1)(x^3 + x^2 + x + 1))/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Thus, lim (x -> 1) = 4.
Correct Answer: A — 0
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Q. Find the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2023)
Solution
The expression can be factored as ((x - 3)(x + 3))/(x - 3). For x ≠ 3, this simplifies to x + 3. Thus, lim (x -> 3) (x + 3) = 6.
Correct Answer: A — 0
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