Q. Calculate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
A.
0
B.
1/2
C.
1
D.
Infinity
Show solution
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1.
Correct Answer:
B
— 1/2
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Q. Calculate the limit: lim (x -> 0) (e^x - 1)/x
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
Using the definition of the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = e^0 = 1.
Correct Answer:
B
— 1
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Q. Calculate the limit: lim (x -> 0) (tan(3x)/x)
A.
3
B.
1
C.
0
D.
Infinity
Show solution
Solution
Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 3, so the limit is 3.
Correct Answer:
A
— 3
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
A.
0
B.
1
C.
2
D.
Undefined
Show solution
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.
Correct Answer:
C
— 2
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)^2
A.
0
B.
1
C.
2
D.
Undefined
Show solution
Solution
Factoring gives (x - 1)(x + 1)/(x - 1)^2 = (x + 1)/(x - 1). Thus, lim (x -> 1) (x + 1)/(x - 1) = 2.
Correct Answer:
C
— 2
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Q. Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
A.
0
B.
1
C.
3
D.
Undefined
Show solution
Solution
Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim (x -> 1) (x^2 + x + 1) = 3.
Correct Answer:
C
— 3
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Q. Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
Factoring gives (x(x - 2))/(x - 2), canceling gives lim (x -> 2) x = 2.
Correct Answer:
D
— Undefined
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Q. Evaluate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
A.
0
B.
1/2
C.
1
D.
Undefined
Show solution
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1/2.
Correct Answer:
B
— 1/2
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Q. Evaluate the limit: lim (x -> 0) (e^x - 1)/x
A.
0
B.
1
C.
e
D.
Infinity
Show solution
Solution
Using L'Hôpital's Rule, we differentiate the numerator and denominator: lim (x -> 0) (e^x)/(1) = e^0 = 1.
Correct Answer:
B
— 1
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Q. Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
Using L'Hôpital's Rule, differentiate the numerator and denominator: lim (x -> 0) (1/(1 + x))/(1) = 1.
Correct Answer:
B
— 1
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Q. Evaluate the limit: lim (x -> 0) (sin(5x)/x)
A.
0
B.
5
C.
1
D.
Infinity
Show solution
Solution
Using the standard limit lim (x -> 0) (sin(x)/x) = 1, we have lim (x -> 0) (sin(5x)/x) = 5 * lim (x -> 0) (sin(5x)/(5x)) = 5 * 1 = 5.
Correct Answer:
B
— 5
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Q. Evaluate the limit: lim (x -> 0) (tan(3x)/x)
A.
0
B.
3
C.
1
D.
Infinity
Show solution
Solution
Using the standard limit lim (x -> 0) (tan(x)/x) = 1, we have lim (x -> 0) (tan(3x)/x) = 3 * lim (x -> 0) (tan(3x)/(3x)) = 3 * 1 = 3.
Correct Answer:
B
— 3
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Q. Evaluate the limit: lim (x -> 0) (x^2 * sin(1/x))
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
Since |sin(1/x)| <= 1, we have |x^2 * sin(1/x)| <= x^2, and thus lim (x -> 0) x^2 * sin(1/x) = 0.
Correct Answer:
A
— 0
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Q. Evaluate the limit: lim (x -> 0) (x^2)/(sin(x))
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
As x approaches 0, sin(x) approaches x, thus lim (x -> 0) (x^2)/(sin(x)) = 0.
Correct Answer:
A
— 0
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Q. Evaluate the limit: lim (x -> ∞) (2x^3 - 3x)/(4x^3 + 5)
Show solution
Solution
Dividing numerator and denominator by x^3 gives lim (x -> ∞) (2 - 3/x^2)/(4 + 5/x^3) = 2/4 = 1/2.
Correct Answer:
B
— 1/2
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Q. Evaluate the limit: lim(x->1) (x^2 - 1)/(x - 1)^2
A.
1
B.
2
C.
0
D.
Undefined
Show solution
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1)^2 = (x+1)/(x-1). Thus, lim(x->1) = 2.
Correct Answer:
B
— 2
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Q. Evaluate the limit: lim(x->infinity) (2x^3 - 3x)/(4x^3 + 5)
A.
1/2
B.
0
C.
1
D.
Infinity
Show solution
Solution
Divide numerator and denominator by x^3: lim(x->infinity) (2 - 3/x^2)/(4 + 5/x^3) = 2/4 = 1/2.
Correct Answer:
A
— 1/2
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Q. Evaluate the limit: lim(x->infinity) (3x^2 + 2)/(5x^2 - 4)
A.
3/5
B.
0
C.
1
D.
Infinity
Show solution
Solution
Divide numerator and denominator by x^2: lim(x->infinity) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer:
A
— 3/5
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Q. Find the limit: lim (x -> 0) (1 - cos(2x))/x^2
A.
0
B.
1
C.
2
D.
Undefined
Show solution
Solution
Using the identity 1 - cos(θ) = 2sin^2(θ/2), we have lim (x -> 0) (1 - cos(2x))/x^2 = lim (x -> 0) (2sin^2(x))/x^2 = 2.
Correct Answer:
C
— 2
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Q. Find the limit: lim (x -> 0) (1 - cos(4x))/(x^2)
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(2x))/(x^2) = 8.
Correct Answer:
B
— 2
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Q. Find the limit: lim (x -> 0) (1 - cos(x))/(x^2)
A.
0
B.
1/2
C.
1
D.
Infinity
Show solution
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1.
Correct Answer:
B
— 1/2
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Q. Find the limit: lim (x -> 0) (cos(x) - 1)/x^2
A.
0
B.
-1/2
C.
1
D.
Infinity
Show solution
Solution
Using the Taylor series expansion for cos(x), we find that lim (x -> 0) (cos(x) - 1)/x^2 = -1/2.
Correct Answer:
B
— -1/2
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Q. Find the limit: lim (x -> 0) (e^x - 1)/x
A.
0
B.
1
C.
e
D.
Undefined
Show solution
Solution
Using the derivative of e^x at x = 0, we find lim (x -> 0) (e^x - 1)/x = 1.
Correct Answer:
B
— 1
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Q. Find the limit: lim (x -> 0) (x^2 * sin(1/x))
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
Since |sin(1/x)| <= 1, we have |x^2 * sin(1/x)| <= |x^2|. As x approaches 0, |x^2| approaches 0, hence the limit is 0.
Correct Answer:
A
— 0
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Q. Find the limit: lim (x -> 0) (x^3)/(e^x - 1)
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
As x approaches 0, e^x - 1 approaches 0. Using L'Hôpital's Rule three times, we find the limit approaches 0.
Correct Answer:
A
— 0
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Q. Find the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
A.
0
B.
1
C.
2
D.
Undefined
Show solution
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.
Correct Answer:
C
— 2
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Q. Find the limit: lim (x -> 1) (x^2 - 1)/(x - 1)^2
A.
0
B.
1
C.
2
D.
Undefined
Show solution
Solution
Factoring gives (x - 1)(x + 1)/(x - 1)^2. Canceling gives lim (x -> 1) (x + 1)/(x - 1) = 2.
Correct Answer:
C
— 2
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Q. Find the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
Show solution
Solution
Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^2 + x + 1) = 3.
Correct Answer:
C
— 3
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Q. Find the limit: lim (x -> 2) (x^2 - 4)/(x - 2)
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
The expression (x^2 - 4)/(x - 2) can be factored as (x - 2)(x + 2)/(x - 2). Canceling (x - 2) gives lim (x -> 2) (x + 2) = 4.
Correct Answer:
C
— 4
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Q. Find the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4)
A.
0
B.
3/5
C.
1
D.
Infinity
Show solution
Solution
Dividing numerator and denominator by x^2, we get lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer:
B
— 3/5
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Showing 1 to 30 of 31 (2 Pages)
Limits MCQ & Objective Questions
Understanding "Limits" is crucial for students preparing for school and competitive exams. This fundamental concept forms the backbone of calculus and is essential for solving various mathematical problems. Practicing MCQs and objective questions on Limits not only enhances your grasp of the topic but also boosts your confidence and scores in exams. Engaging with these practice questions helps you identify important questions and refine your exam preparation strategy.
What You Will Practise Here
Definition and basic concepts of Limits
Types of Limits: Finite, Infinite, and One-Sided Limits
Limit laws and properties
Evaluating Limits using substitution and factoring
Understanding continuity and its relation to Limits
Applications of Limits in real-world problems
Graphical interpretation of Limits
Exam Relevance
The topic of Limits is frequently tested in various examinations, including CBSE, State Boards, NEET, and JEE. Students can expect questions that require them to evaluate Limits, apply limit laws, and solve problems involving continuity. Common question patterns include multiple-choice questions that assess both conceptual understanding and problem-solving skills, making it essential to practice Limits MCQ questions thoroughly.
Common Mistakes Students Make
Confusing one-sided Limits with two-sided Limits
Overlooking the importance of continuity when evaluating Limits
Misapplying limit laws, especially in complex expressions
Neglecting to check for indeterminate forms before solving
Failing to interpret graphical representations of Limits accurately
FAQs
Question: What are the basic types of Limits I should know?Answer: You should be familiar with finite Limits, infinite Limits, and one-sided Limits, as they are fundamental to understanding the concept.
Question: How can I effectively prepare for Limits questions in exams?Answer: Regular practice of Limits objective questions with answers and reviewing common mistakes can significantly improve your understanding and performance.
Ready to master Limits? Dive into our practice MCQs and test your understanding today! The more you practice, the better prepared you'll be for your exams.
