Limits – Introduction MCQs With Answer

Limits – Introduction MCQs With Answer provides B. Pharm students a concise, exam-oriented review of limit concepts used in pharmacokinetics and pharmaceutical calculations. This introduction covers one-sided limits, limits at infinity, continuity, indeterminate forms, squeeze theorem, and L’Hôpital’s rule with practical examples like drug concentration as t→0 or t→∞ and dose-rate behavior. These MCQs reinforce analytical skills for solving rational, trigonometric, exponential, and logarithmic limits commonly seen in clinical and formulation problems. Answers and concise explanations help rapid revision. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the definition of the limit of f(x) as x approaches a?

• The value f(x) takes at x = a only
• The value f(x) approaches as x gets arbitrarily close to a
• The average of left and right hand values at a
• The derivative of f at x = a

Correct Answer: The value f(x) approaches as x gets arbitrarily close to a

Q2. If lim(x→a−) f(x) = L and lim(x→a+) f(x) = L, what can be concluded?

• Function has an infinite discontinuity at a
• Two-sided limit lim(x→a) f(x) = L
• Function is differentiable at a
• Function has a removable discontinuity at a

Correct Answer: Two-sided limit lim(x→a) f(x) = L

Q3. Evaluate lim(x→2) (x^2 − 4)/(x − 2).

• 0
• 2
• 4
• Undefined

Correct Answer: 4

Q4. For the limit lim(x→1) (x^3 − 1)/(x − 1), which method is best?

• Use L’Hôpital’s rule directly
• Factor numerator and cancel (x − 1)
• Apply squeeze theorem
• Approximate numerically only

Correct Answer: Factor numerator and cancel (x − 1)

Q5. Evaluate lim(x→0) sin x / x (x in radians).

• 0
• 1
• Undefined
• ∞

Correct Answer: 1

Q6. Evaluate lim(x→0) (1 − cos x)/x^2.

• 0
• 1/2
• 1
• ∞

Correct Answer: 1/2

Q7. Evaluate lim(x→0) (e^x − 1)/x.

• 0
• 1
• e
• ∞

Correct Answer: 1

Q8. Which of the following is an indeterminate form?

• 0/0
• 1/2
• 3
• −1

Correct Answer: 0/0

Q9. L’Hôpital’s rule can be applied to which types of limits?

• Only 0×∞ forms
• Only finite non-zero limits
• 0/0 and ∞/∞ indeterminate forms
• Any limit regardless of form

Correct Answer: 0/0 and ∞/∞ indeterminate forms

Q10. If degrees of numerator and denominator are equal in a rational function, lim(x→∞) equals:

• 0
• Ratio of leading coefficients
• ∞
• Does not exist

Correct Answer: Ratio of leading coefficients

Q11. The squeeze (sandwich) theorem is most useful when:

• You can bound a function between two functions with the same limit
• Limits involve only polynomials
• L’Hôpital’s rule is applicable
• Function is discontinuous everywhere

Correct Answer: You can bound a function between two functions with the same limit

Q12. A function f is continuous at a if:

• lim(x→a) f(x) exists and equals f(a)
• Only left-hand limit exists
• Only right-hand limit exists
• f has a vertical asymptote at a

Correct Answer: lim(x→a) f(x) exists and equals f(a)

Q13. Which limit indicates a removable discontinuity at x = a?

• lim(x→a) f(x) exists but f(a) is different or undefined
• Left and right limits are different
• Limit is infinite
• No limit exists due to oscillation

Correct Answer: lim(x→a) f(x) exists but f(a) is different or undefined

Q14. If lim(x→a) f(x) = ∞, what does this indicate?

• Removable discontinuity
• Vertical asymptote at x = a
• Function is continuous at a
• Two-sided finite limit exists

Correct Answer: Vertical asymptote at x = a

Q15. For the exponential decay C(t) = C0 e^(−kt), lim(t→∞) C(t) equals:

• C0
• k
• 0
• ∞

Correct Answer: 0

Q16. For the same C(t) = C0 e^(−kt), lim(t→0) C(t) equals:

• 0
• C0
• k
• Undefined

Correct Answer: C0

Q17. Evaluate lim(n→∞) (1 + 1/n)^n.

• 0
• 1
• e
• ∞

Correct Answer: e

Q18. Evaluate lim(n→∞) n/(n + 1).

• 0
• 1/2
• 1
• ∞

Correct Answer: 1

Q19. Evaluate lim(x→0) ln(1 + x)/x.

• 0
• 1
• Undefined
• ∞

Correct Answer: 1

Q20. Evaluate lim(x→0) sin(2x)/x.

• 0
• 1
• 2
• Undefined

Correct Answer: 2

Q21. Evaluate lim(x→∞) arctan x.

• 0
• π/2
• −π/2
• Does not exist

Correct Answer: π/2

Q22. Evaluate lim(x→∞) (√(x + 1) − √x).

• ∞
• 1
• 0
• 1/2

Correct Answer: 0

Q23. Evaluate lim(x→a) (√x − √a)/(x − a) using conjugate technique.

• 0
• 1/(2√a)
• 2√a
• Undefined

Correct Answer: 1/(2√a)

Q24. The derivative f'(a) is defined as which limit?

• lim(Δx→0) [f(a + Δx) − f(a)]
• lim(Δx→0) [f(a + Δx) − f(a)]/Δx
• lim(Δx→0) f(a + Δx)*Δx
• lim(Δx→0) f(a)/Δx

Correct Answer: lim(Δx→0) [f(a + Δx) − f(a)]/Δx

Q25. Evaluate lim(x→0) x·sin(1/x).

• 1
• 0
• Does not exist due to oscillation
• ∞

Correct Answer: 0

Q26. Does lim(x→0) sin(1/x) exist?

• Yes, equals 0
• Yes, equals 1
• No, it oscillates between −1 and 1
• Yes, equals ∞

Correct Answer: No, it oscillates between −1 and 1

Q27. Evaluate lim(n→∞) 1/n^2 for a sequence.

• 0
• 1
• ∞
• Does not exist

Correct Answer: 0

Q28. Evaluate lim(x→∞) e^{kx}/x for k > 0.

• 0
• 1
• ∞
• Depends on k

Correct Answer: ∞

Q29. Evaluate lim(x→0+) ln x.

• 0
• 1
• −∞
• ∞

Correct Answer: −∞

Q30. Evaluate lim(x→0) tan x / x.

• 0
• 1
• Undefined
• ∞

Correct Answer: 1

Q31. Which limit gives the leading second-order behavior of cosine?

• lim(x→0) (1 − cos x)/x
• lim(x→0) (1 − cos x)/x^2
• lim(x→0) sin x / x
• lim(x→0) cos x

Correct Answer: lim(x→0) (1 − cos x)/x^2

Q32. Evaluate lim(x→∞) ln x / x.

• ∞
• 1
• 0
• Does not exist

Correct Answer: 0

Q33. Evaluate lim(x→1) (x^2 − 1)/(x − 1).

• 0
• 1
• 2
• Undefined

Correct Answer: 2

Q34. Evaluate lim(x→2) (x^3 − 8)/(x − 2).

• 6
• 8
• 12
• Undefined

Correct Answer: 12

Q35. The limit lim(x→0) sin x / x equals 1. This result depends on:

• Angle measured in degrees
• Angle measured in radians
• Base of logarithm used
• Value of π being 2

Correct Answer: Angle measured in radians

Q36. Evaluate lim(n→∞) (ln n)/n.

• 0
• 1
• ∞
• Does not exist

Correct Answer: 0

Q37. For a geometric sequence r^n with |r| < 1, lim(n→∞) r^n equals:

• 1
• r
• 0
• Does not exist

Correct Answer: 0

Q38. Which condition is required for continuity at x = a?

• Left-hand limit exists but right-hand does not
• Both one-sided limits exist and equal f(a)
• Only f(a) exists
• Derivative at a exists

Correct Answer: Both one-sided limits exist and equal f(a)

Q39. Evaluate lim(x→0) sin x / (1 − cos x).

• 0
• 1
• 2
• ∞ (diverges to infinity)

Correct Answer: ∞ (diverges to infinity)

Q40. Evaluate lim(x→0+) x ln x.

• 0
• −∞
• 1
• ∞

Correct Answer: 0

Q41. Evaluate lim(x→0) (1 + x)^{1/x}.

• 0
• 1
• e
• ∞

Correct Answer: e

Q42. Evaluate lim(x→0) (sin x − x)/x^3.

• 0
• −1/6
• 1/6
• Does not exist

Correct Answer: −1/6

Q43. Evaluate lim(x→∞) x^2 / e^x.

• 0
• 1
• ∞
• Does not exist

Correct Answer: 0

Q44. For large x, compare e^x and x^n. lim(x→∞) e^x / x^n equals:

• 0
• 1
• ∞
• Depends on n

Correct Answer: ∞

Q45. Evaluate lim(x→∞) (1 + a/x)^x where a is constant.

• 1 + a
• e^a
• a
• Does not exist

Correct Answer: e^a

Q46. Evaluate lim(x→∞) (√(x^2 + x) − x).

• 0
• 1/2
• ∞
• 1

Correct Answer: 1/2

Q47. Evaluate lim(n→∞) (1 − 1/n)^n.

• 0
• 1
• 1/e
• e

Correct Answer: 1/e

Q48. Evaluate lim(x→∞) (3x^2 + 2x)/(6x^2 − x).

• 0
• 1/2
• 3/6
• ∞

Correct Answer: 1/2

Q49. Consider f(x) = x/|x|. Does lim(x→0) f(x) exist?

• Yes, equals 0
• Yes, equals 1
• No, left and right limits differ
• Yes, equals −1

Correct Answer: No, left and right limits differ

Q50. Evaluate lim(x→0) (e^x − 1 − x)/x^2.

• 0
• 1/2
• 1
• ∞

Correct Answer: 1/2

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