Limit of sinθ / θ = 1 as θ→0 MCQs With Answer

Introduction: The limit of sinθ/θ = 1 as θ→0 is a fundamental trigonometric limit every B.Pharm student should master for calculus applications in pharmaceutical modeling, pharmacokinetics, and signal analysis. Understanding this trigonometric limit, why radians are essential, and proofs using the squeeze theorem, Taylor series, or L’Hospital’s rule strengthens your problem-solving skills. This topic connects small-angle approximation, derivative of sin x at zero, and error estimates — all useful when linearizing dose-response curves or oscillatory systems in drug delivery. Clear familiarity with this limit helps in simplifying complex expressions and improves accuracy in analytical derivations. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the value of the limit lim_{θ→0} (sinθ)/θ?

• 0
• 1
• Infinity
• Does not exist

Correct Answer: 1

Q2. Why must θ be measured in radians for the limit lim_{θ→0} sinθ/θ = 1 to hold directly?

• Degrees give the same result but are less common
• Radians make the series expansion coefficients match geometric derivatives
• Because calculators use radians only
• It only holds for integer multiples of π

Correct Answer: Radians make the series expansion coefficients match geometric derivatives

Q3. Which theorem provides a classical geometric proof that lim_{θ→0} sinθ/θ = 1?

• Mean Value Theorem
• Squeeze (Sandwich) Theorem
• Intermediate Value Theorem
• Fundamental Theorem of Algebra

Correct Answer: Squeeze (Sandwich) Theorem

Q4. Using Taylor series, sinθ = θ – θ^3/6 + …, what does this series indicate about sinθ/θ as θ→0?

• sinθ/θ → 0
• sinθ/θ → 1 with error O(θ^2)
• sinθ/θ oscillates
• sinθ/θ → ∞

Correct Answer: sinθ/θ → 1 with error O(θ^2)

Q5. Evaluate lim_{θ→0} (sin2θ)/θ.

• 1
• 2
• 0
• Does not exist

Correct Answer: 2

Q6. If f(θ)=sinθ, what is f'(0) using the limit definition of derivative?

• 0
• 1
• -1
• Undefined

Correct Answer: 1

Q7. Which limit is equivalent to lim_{θ→0} sinθ/θ by substitution u=θ/2?

• lim_{u→0} (sin2u)/2u
• lim_{u→0} (sin u)/2u
• lim_{u→0} (sin u)/u^2
• lim_{u→0} (sin2u)/u^2

Correct Answer: lim_{u→0} (sin2u)/2u

Q8. Using L’Hospital’s rule, how do you evaluate lim_{θ→0} sinθ/θ?

• Differentiate numerator and denominator to get cosθ/1 and evaluate at 0
• Apply L’Hospital once to get -cosθ/-1
• L’Hospital cannot be used here
• Differentiate twice and evaluate

Correct Answer: Differentiate numerator and denominator to get cosθ/1 and evaluate at 0

Q9. What is lim_{θ→0} tanθ/θ?

• 0
• 1
• Undefined
• π/2

Correct Answer: 1

Q10. For small θ, sinθ ≈ θ. What is the leading order relative error term?

• O(θ)
• O(θ^2)
• O(θ^3)
• O(1/θ)

Correct Answer: O(θ^2)

Q11. Evaluate lim_{θ→0} (1 – cosθ)/θ^2.

• 0
• 1/2
• 1
• Infinity

Correct Answer: 1/2

Q12. Which of the following justifies sinθ < θ for θ in (0, π/2)?

• Concavity of sine on (0, π/2)
• Monotonicity of sine
• Symmetry of sine
• Periodicity of sine

Correct Answer: Concavity of sine on (0, π/2)

Q13. If θ is in degrees, what happens to sinθ/θ as θ→0 (with θ in degrees)?

• Approaches 1 without change
• Approaches π/180
• Approaches 180/π
• Diverges to infinity

Correct Answer: Approaches π/180

Q14. Consider lim_{θ→0} (sin3θ)/(sin2θ). What is the limit?

• 3/2
• 2/3
• 1
• 0

Correct Answer: 3/2

Q15. Using series, what is lim_{θ→0} (sinθ – θ)/θ^3?

• 0
• -1/6
• 1/6
• Infinity

Correct Answer: -1/6

Q16. Which statement is true about sinθ/θ for θ near 0?

• It increases without bound
• It oscillates between -1 and 1
• It approaches 1 smoothly from below for positive θ
• It is undefined for small θ

Correct Answer: It approaches 1 smoothly from below for positive θ

Q17. For small-angle linearization in pharmacokinetic modeling, sinθ ≈ θ is valid when θ is expressed in:

• Degrees
• Radians
• Gradians
• Any unit with a conversion factor applied afterward

Correct Answer: Radians

Q18. Evaluate lim_{θ→0} (sinθ)/(2θ) .

• 1
• 1/2
• 2
• 0

Correct Answer: 1/2

Q19. Which limit equals 1 by using sinθ/θ = 1 as θ→0?

• lim_{θ→0} (sin(θ^2))/θ
• lim_{θ→0} (sin(θ))/2θ
• lim_{θ→0} (sin(θ)/θ)^{1/θ}
• lim_{θ→0} (sin(kθ))/(kθ) for constant k

Correct Answer: lim_{θ→0} (sin(kθ))/(kθ) for constant k

Q20. If g(θ)=sinθ/θ for θ≠0 and g(0)=1, what is true about g at θ=0?

• g is discontinuous at 0
• g has a removable discontinuity at 0
• g has a jump discontinuity at 0
• g is not defined at 0 even after extension

Correct Answer: g has a removable discontinuity at 0

Q21. Which small-angle identity derives directly from lim_{θ→0} sinθ/θ = 1?

• sinθ ≈ θ
• cosθ ≈ θ
• tanθ ≈ θ^2
• sinθ ≈ θ^2

Correct Answer: sinθ ≈ θ

Q22. For which function h(θ) does lim_{θ→0} h(θ) = 1 follow from lim_{θ→0} sinθ/θ = 1?

• h(θ) = (sinθ)/tanθ
• h(θ) = (1 – cosθ)/θ
• h(θ) = (sin2θ)/2θ
• h(θ) = cosθ/θ

Correct Answer: h(θ) = (sin2θ)/2θ

Q23. What is lim_{θ→0} (sinθ)/(θ + θ^3)?

• 1
• 0
• Infinite
• 1 with leading correction -θ^2

Correct Answer: 1 with leading correction -θ^2

Q24. The inequality sinθ < θ < tanθ for θ in (0, π/2) helps prove which result?

• sinθ/θ → 0
• sinθ/θ oscillates
• sinθ/θ → 1
• tanθ/θ → 0

Correct Answer: sinθ/θ → 1

Q25. Evaluate lim_{θ→0} (sinθ)/√θ.

• 0
• 1
• Infinity
• Undefined

Correct Answer: 0

Q26. For small θ, which approximation is more accurate to second order?

• sinθ ≈ θ – θ^2/2
• sinθ ≈ θ – θ^3/6
• sinθ ≈ θ + θ^3/6
• sinθ ≈ θ^2

Correct Answer: sinθ ≈ θ – θ^3/6

Q27. If lim_{θ→0} sinθ/θ = 1, what is lim_{θ→0} (sinθ)/|θ| ?

• 1
• -1
• Undefined because sign changes
• 0

Correct Answer: 1

Q28. Which limit uses sinθ/θ to evaluate lim_{θ→0} (sin(θ) + θ)/θ ?

• 2
• 1
• 0
• Infinity

Correct Answer: 2

Q29. Evaluate lim_{θ→0} (sinθ – θ)/θ.

• -1
• 0
• -1/6
• 1

Correct Answer: 0

Q30. Which trigonometric limit is directly derived from the derivative of sin x at x=0?

• lim_{θ→0} (1 – cosθ)/θ
• lim_{θ→0} sinθ/θ
• lim_{θ→0} cosθ/θ
• lim_{θ→0} tanθ/θ^2

Correct Answer: lim_{θ→0} sinθ/θ

Q31. If θ_n → 0, which sequence limit holds true?

• sinθ_n/θ_n → 0
• sinθ_n/θ_n → 1
• sinθ_n/θ_n is unbounded
• sinθ_n/θ_n alternates

Correct Answer: sinθ_n/θ_n → 1

Q32. Evaluate lim_{θ→0} (sinθ)/(θ – θ^3/6).

• 0
• 1
• 6
• Undefined

Correct Answer: 1

Q33. For small θ, which is an appropriate linear model replacement used in dose-response linearization?

• sinθ ≈ 1
• sinθ ≈ θ
• sinθ ≈ θ^2/2
• tanθ ≈ θ^2

Correct Answer: sinθ ≈ θ

Q34. Which limit equals 1 by direct substitution after factoring θ?

• lim_{θ→0} (sinθ)/(θ)
• lim_{θ→0} (sinθ)/(θ^2)
• lim_{θ→0} (sinθ)/(θ^3)
• lim_{θ→0} (sinθ – θ)/(θ)

Correct Answer: lim_{θ→0} (sinθ)/(θ)

Q35. The error term in sinθ expansion is dominated by which power near zero?

• θ
• θ^2
• θ^3
• θ^4

Correct Answer: θ^3

Q36. Evaluate lim_{θ→0} (sinθ)/(θ + o(θ)).

• 0
• 1
• Depends on o(θ)
• Infinity

Correct Answer: 1

Q37. If one uses the approximation sinθ ≈ θ for modeling, what must be small to ensure accuracy?

• θ measured in degrees
• θ value (in radians)
• θ inverse
• Only the sign of θ

Correct Answer: θ value (in radians)

Q38. Evaluate lim_{θ→0} (sinθ)/(arctanθ).

• 0
• 1
• π/4
• Undefined

Correct Answer: 1

Q39. Which substitution directly reduces lim_{θ→0} (sin(aθ))/(bθ) to a known form?

• u = aθ
• u = θ/b
• u = θ^2
• u = sinθ

Correct Answer: u = aθ

Q40. Evaluate lim_{θ→0} (sinθ – θ + θ^3/6)/θ^5.

• 0
• -1/120
• 1/120
• Infinity

Correct Answer: 0

Q41. For a continuous function φ with φ(0)=0 and φ'(0)=1, which limit mirrors sinθ/θ?

• lim_{θ→0} φ(θ)/θ = 1
• lim_{θ→0} φ(θ)/θ = 0
• lim_{θ→0} φ(θ)/θ = φ(0)
• lim_{θ→0} φ(θ) = 1

Correct Answer: lim_{θ→0} φ(θ)/θ = 1

Q42. If lim_{θ→0} sinθ/θ = 1, what is lim_{θ→0} (1 – cosθ)/θ ?

• 0
• 1/2
• Infinity
• 1

Correct Answer: 0

Q43. Which limit evaluates to 1 using sinθ/θ and properties of limits?

• lim_{θ→0} (sin^2θ)/θ^2
• lim_{θ→0} (sinθ)/θ^2
• lim_{θ→0} (sinθ)^3/θ
• lim_{θ→0} sinθ/(θ^4)

Correct Answer: lim_{θ→0} (sin^2θ)/θ^2

Q44. Evaluate lim_{θ→0} (sinθ)/(θ) * (1 + θ^2).

• 0
• 1
• Depends on θ
• Infinity

Correct Answer: 1

Q45. If sinθ/θ → 1, what is lim_{θ→0} (sin(θ + θ^2) / (θ + θ^2))?

• 0
• 1
• Depends on higher terms
• 2

Correct Answer: 1

Q46. Which limit follows directly: lim_{θ→0} (sin3θ)/(3θ) = ?

• 1/3
• 1
• 3
• 0

Correct Answer: 1

Q47. Evaluate lim_{θ→0} (sinθ)/sin(2θ) .

• 1/2
• 2
• 1
• 0

Correct Answer: 1/2

Q48. For a small perturbation used in model linearization, which is true?

• sinθ is linear for large θ
• The linear term of sinθ expansion is θ, so linearization uses θ
• Quadratic term dominates near zero
• sinθ ≈ constant near zero

Correct Answer: The linear term of sinθ expansion is θ, so linearization uses θ

Q49. Which limit equals 1 by combining sinθ/θ with a constant multiplier?

• lim_{θ→0} (2 sinθ)/(2θ)
• lim_{θ→0} (2 sinθ)/θ
• lim_{θ→0} sinθ/(2θ)
• lim_{θ→0} (sinθ)/(0.5θ)

Correct Answer: lim_{θ→0} (2 sinθ)/(2θ)

Q50. What important calculus fact for B.Pharm students relates directly to lim_{θ→0} sinθ/θ = 1?

• The derivative of sin x at 0 equals π
• The derivative of sin x at 0 equals 1
• sin x is not differentiable at 0
• The integral of sin x diverges at 0

Correct Answer: The derivative of sin x at 0 equals 1

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