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
- −am
- 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)=xm, 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) xr equals:
- ap/q if defined
- Always 0
- Does not exist
- −ap/q
Correct Answer: ap/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 + xm] = an + am?
- 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) x1/2 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)/(xm) equals an−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 x1/3 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) xp/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) x1/3, 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 xk, lim(x→a) P(x) equals:
- Σ c_k ak
- 0
- Only leading coefficient matters
- Infinite
Correct Answer: Σ c_k ak
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
