Limits & Continuity

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Q1. Find the limit: lim (x -> 2) (x^2 + 3x - 10)/(x - 2) (2021)

Solution:

Factoring gives (x - 2)(x + 5)/(x - 2). For x ≠ 2, this simplifies to x + 5. Evaluating at x = 2 gives 7.

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Q2. Find the limit: lim (x -> 2) (x^2 - 3x + 2)/(x - 2) (2021)

Solution:

The expression is undefined at x=2. The limit does not exist as the function approaches infinity.

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Q3. Determine the limit: lim (x -> 1) (x^3 - 1)/(x - 1) (2020)

Solution:

Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^2 + x + 1. Evaluating at x = 1 gives 3.

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Q4. Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)

Solution:

Factoring gives (x - 3)(x + 3)/(x - 3). Canceling (x - 3) gives lim (x -> 3) (x + 3) = 6.

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Q5. Evaluate the limit: lim (x -> 0) (tan(x)/x) (2023)

Solution:

Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.

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Q6. 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.

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Q7. Evaluate the limit: lim (x -> 0) (x^3)/(sin(x)) (2022)

Solution:

As x approaches 0, x^3 approaches 0 and sin(x) approaches 0, thus the limit is 0.

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Q8. 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.

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Q9. Find the limit: lim (x -> 0) (x^2)/(sin(x)) (2023)

Solution:

As x approaches 0, sin(x) approaches x, thus lim (x -> 0) (x^2/sin(x)) = lim (x -> 0) (x^2/x) = lim (x -> 0) x = 0.

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Q10. Determine the limit: lim (x -> 0) (tan(5x)/x) (2022)

Solution:

Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 5. Thus, lim (x -> 0) (tan(5x)/x) = 5.