Limit of xn at x→a MCQs With Answer

Introduction: Understanding the Limit of xⁿ at x→a is essential for B.Pharm students studying calculus applied to pharmaceutical calculations. This topic covers how power functions behave as the variable approaches a point, continuity of polynomials, and special cases like negative or fractional exponents and root functions. Mastery of limit laws, substitution, and how limits relate to derivatives (for example (xⁿ−aⁿ)/(x−a)) is useful in dose-response modeling and kinetics. This Student-friendly guide focuses on Limit of xⁿ at x→a MCQs With Answer to strengthen concept clarity, problem-solving speed, and exam readiness. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is lim(x→a) xⁿ for a real number a and integer n≥0?

• a
• aⁿ
• n·a
• 0

Correct Answer: aⁿ

Q2. For n a positive integer, which property justifies lim(x→a) xⁿ = aⁿ?

• Intermediate Value Theorem
• Continuity of polynomial functions
• Mean Value Theorem
• Bolzano’s theorem

Correct Answer: Continuity of polynomial functions

Q3. If n is negative (n = −m) and a ≠ 0, lim(x→a) xⁿ equals:

• 0
• a^−m
• −a^m
• Does not exist

Correct Answer: a^−m

Q4. lim(x→0) x³ equals:

• 0
• 1
• Undefined
• 3

Correct Answer: 0

Q5. If f(x)=xⁿ and g(x)=x^m, what is lim(x→a) [f(x)·g(x)]?

• lim f(x) + lim g(x)
• lim f(x) · lim g(x)
• lim f(x) / lim g(x)
• The product does not have a limit

Correct Answer: lim f(x) · lim g(x)

Q6. For even n, the function xⁿ near x=a preserves which of the following?

• Sign of x
• Non-negativity for all real x
• Odd symmetry
• Periodicity

Correct Answer: Non-negativity for all real x

Q7. Evaluate lim(x→2) (x⁴).

• 8
• 16
• 4
• 256

Correct Answer: 16

Q8. For rational exponent r=p/q with q odd, lim(x→a) x^r equals:

• a^p/q if defined
• Always 0
• Does not exist
• −a^p/q

Correct Answer: a^p/q if defined

Q9. If a=0 and n is negative, lim(x→0) xⁿ is:

• 0
• ∞ or does not exist
• 1
• −∞

Correct Answer: ∞ or does not exist

Q10. Which limit rule allows lim(x→a) [xⁿ + x^m] = aⁿ + a^m?

• Sum rule for limits
• Quotient rule for limits
• Chain rule
• Integration rule

Correct Answer: Sum rule for limits

Q11. lim(x→a) (xⁿ − aⁿ)/(x−a) equals:

• n·aⁿ⁻¹
• aⁿ
• 0
• 1

Correct Answer: n·aⁿ⁻¹

Q12. The expression lim(x→a) x^1/2 requires what condition on a for real-valued limit?

• a ≥ 0
• a ≤ 0
• a ≠ 1
• All real a

Correct Answer: a ≥ 0

Q13. For continuous function f(x)=xⁿ, what is lim(x→a+) f(x) compared to lim(x→a−) f(x)?

• They are equal
• Right-hand is always larger
• Left-hand is always larger
• They differ unless n is even

Correct Answer: They are equal

Q14. lim(x→∞) xⁿ for n>0 is:

• 0
• ∞
• 1
• Does not exist because oscillatory

Correct Answer: ∞

Q15. If f(x)=xⁿ and a is nonzero, continuity implies which immediate result?

• Derivative does not exist
• lim(x→a) f(x) = f(a)
• f is bounded everywhere
• f has a removable discontinuity

Correct Answer: lim(x→a) f(x) = f(a)

Q16. Which technique directly gives lim(x→a) xⁿ = aⁿ by substituting x=a?

• Direct substitution using continuity
• Partial fractions
• Integration by parts
• Squeeze theorem

Correct Answer: Direct substitution using continuity

Q17. lim(x→a) (x² − a²)/(x−a) equals:

• 2a
• a²
• a
• 0

Correct Answer: 2a

Q18. For x near a, xⁿ − aⁿ can be factored using:

• Difference of powers formula
• Quadratic formula
• Binomial theorem only for n=2
• Taylor series only

Correct Answer: Difference of powers formula

Q19. If a=−2 and n is odd, lim(x→−2) xⁿ equals:

• −2
• (−2)ⁿ
• 2ⁿ
• Does not exist

Correct Answer: (−2)ⁿ

Q20. Which statement is true for lim(x→a) xⁿ when n is even?

• Limit equals aⁿ and is non-negative
• Limit flips sign at a
• Limit is undefined for negative a
• Limit depends on direction only

Correct Answer: Limit equals aⁿ and is non-negative

Q21. lim(x→a) (x³ + 2x² − x) equals:

• a³ + 2a² − a
• a⁶
• 3a² + 4a − 1
• Does not exist

Correct Answer: a³ + 2a² − a

Q22. To prove lim(x→a) xⁿ = aⁿ using epsilon-delta, you need to bound |xⁿ−aⁿ| by:

• n·|x−a|·M for some M
• |x−a|
• Integers only
• Infinity

Correct Answer: n·|x−a|·M for some M

Q23. If a=0 and n>0, lim(x→0) xⁿ equals 0 by which reasoning?

• Because xⁿ→0 as x→0 for n>0
• Because derivative at 0 is 0
• Because integral from 0 to a is 0
• It does not approach 0

Correct Answer: Because xⁿ→0 as x→0 for n>0

Q24. For limit involving (xⁿ − aⁿ)/(x−a), which calculus concept does this limit represent?

• Second derivative
• Derivative of xⁿ at a
• Integral of xⁿ
• Mean value of xⁿ

Correct Answer: Derivative of xⁿ at a

Q25. lim(x→a) (xⁿ)/(x^m) equals a^n−m provided:

• a ≠ 0 if m>n
• m = n only
• a = 0 always
• The limit is always infinite

Correct Answer: a ≠ 0 if m>n

Q26. For sequence x_k → a, what is lim(k→∞) x_kⁿ?

• Depends on sequence only
• aⁿ
• n·a
• 0

Correct Answer: aⁿ

Q27. Which is a correct limit when approaching from the right for x^1/3 at a negative a?

• Different from left-hand limit
• Equal to real cube root of a
• Undefined because negative
• Infinite

Correct Answer: Equal to real cube root of a

Q28. lim(x→a) [xⁿ − aⁿ]/(x−a) for n=1 equals:

• 1
• a
• 0
• Does not exist

Correct Answer: 1

Q29. If lim(x→a) f(x) = L and f(x)=xⁿ, then L equals:

• 0
• aⁿ
• n·a
• Undefined

Correct Answer: aⁿ

Q30. For small h, (a+h)ⁿ ≈ aⁿ + n aⁿ⁻¹ h follows from which concept?

• Continuity fails
• Linear approximation / derivative
• Integration
• Riemann sum

Correct Answer: Linear approximation / derivative

Q31. Evaluate lim(x→−1) x².

• −1
• 1
• 0
• 2

Correct Answer: 1

Q32. For non-integer rational exponent p/q with even q, lim(x→a) x^p/q exists for real values only if:

• a ≥ 0 for real principal root
• a ≤ 0
• a ≠ 0
• Always exists

Correct Answer: a ≥ 0 for real principal root

Q33. Which of the following is true about lim(x→a) xⁿ when a→0 and n→∞ simultaneously? (Consider fixed x variable approaching 0 first)

• Indeterminate without order of limits
• Always 0
• Always ∞
• Equals 1

Correct Answer: Indeterminate without order of limits

Q34. The squeeze theorem is useful for limits of xⁿ when:

• You can bound xⁿ between two functions with known limits
• Only for polynomials of degree 1
• Never useful
• Only for rational functions

Correct Answer: You can bound xⁿ between two functions with known limits

Q35. lim(x→a) |x|ⁿ equals:

• |a|ⁿ
• aⁿ always
• −|a|ⁿ
• Depends on direction

Correct Answer: |a|ⁿ

Q36. If a=0 and n=0 (x⁰ conventionally 1), lim(x→0) x⁰ is:

• 0
• 1
• Undefined due to 0⁰
• Depends on path

Correct Answer: Undefined due to 0⁰

Q37. lim(x→a) (xⁿ − aⁿ)/(x−a) can be computed by factorization giving:

• Sum of geometric-like terms: aⁿ⁻¹ + aⁿ⁻²x + … + xⁿ⁻¹
• Only a single term
• Infinity always
• Zero always

Correct Answer: Sum of geometric-like terms: aⁿ⁻¹ + aⁿ⁻²x + … + xⁿ⁻¹

Q38. For small x near a, continuity of xⁿ implies what about error when substituting?

• Error can be made arbitrarily small
• Error is fixed and large
• Error increases without bound
• Substitution is invalid

Correct Answer: Error can be made arbitrarily small

Q39. lim(x→a) (xⁿ/xⁿ) for x≠0 equals:

• 0
• 1
• a
• Does not exist

Correct Answer: 1

Q40. If f(x)=xⁿ and n is an integer, then f is continuous at which points?

• All real numbers
• Only at x=0
• Only positive x
• Only integers

Correct Answer: All real numbers

Q41. lim(x→a) xⁿ where n is even and a is negative gives:

• Negative result
• Positive result equal to aⁿ
• Undefined
• Zero

Correct Answer: Positive result equal to aⁿ

Q42. Evaluate lim(x→3) (x⁰).

• 3
• 0
• 1 for x≠0
• Undefined

Correct Answer: 1 for x≠0

Q43. In pharmacokinetic modeling, why is continuity of xⁿ important when using limits?

• Ensures model predictions change smoothly with parameters
• Makes equations nonlinear always
• Prevents differentiation
• Causes discontinuities in concentration

Correct Answer: Ensures model predictions change smoothly with parameters

Q44. lim(x→a) (x·xⁿ) equals:

• aⁿ⁺¹
• aⁿ⁻¹
• a
• 0

Correct Answer: aⁿ⁺¹

Q45. If lim(x→a) xⁿ = aⁿ, then for continuous g, lim(x→a) g(xⁿ) =:

• g(aⁿ)
• g(n)
• Cannot determine
• 0

Correct Answer: g(aⁿ)

Q46. For limit lim(x→0) x^1/3, the value is:

• 0
• Undefined because root
• 1
• ∞

Correct Answer: 0

Q47. lim(x→a) (xⁿ − aⁿ)/(x−a) can also be evaluated using which theorem?

• Mean Value Theorem (MVT)
• Fundamental Theorem of Algebra
• Intermediate Value Theorem only
• Green’s theorem

Correct Answer: Mean Value Theorem (MVT)

Q48. For a polynomial P(x)=Σ c_k x^k, lim(x→a) P(x) equals:

• Σ c_k a^k
• 0
• Only leading coefficient matters
• Infinite

Correct Answer: Σ c_k a^k

Q49. Which of the following is true about lim(x→a) xⁿ when considering complex a and n integer?

• Limit equals aⁿ using complex continuity
• Limit does not exist in complex plane
• Only real limits allowed
• Limit equals conjugate of aⁿ

Correct Answer: Limit equals aⁿ using complex continuity

Q50. When applying limits in B.Pharm calculations, which practical tip about xⁿ limits is most useful?

• Direct substitution is valid when function is continuous at a
• Always use l’Hôpital’s rule
• Avoid substitution to prevent errors
• Limits are irrelevant in pharmaceutical modeling

Correct Answer: Direct substitution is valid when function is continuous at a

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