Limits & Continuity

⏱ Time: 0s

Q1. Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

Solution:

Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^3 + x^2 + x + 1) = 4.

⏱ Time: 0s

Q2. Calculate the limit: lim (x -> 0) (tan(5x)/x) (2022)

Solution:

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

⏱ Time: 0s

Q3. Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)

Solution:

Using the Taylor series expansion for sin(x), we find that lim (x -> 0) (x - sin(x))/x^3 = 1/6.

⏱ Time: 0s

Q4. Determine the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

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. Evaluating at x = 1 gives 4.

⏱ Time: 0s

Q5. 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.

⏱ Time: 0s

Q6. 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.

⏱ Time: 0s

Q7. Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)

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.

⏱ Time: 0s

Q8. 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.

⏱ Time: 0s

Q9. 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.

⏱ Time: 0s

Q10. 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.