Limits & Continuity: Essential Concepts for Competitive Exams

Limits & Continuity

Q. Calculate the limit: lim (x -> 0) (tan(5x)/x) (2022)

A. 0
B. 1
C. 5
D. Undefined

Solution

Using the limit property lim (x -> 0) (tan(kx)/x) = k, we have lim (x -> 0) (tan(5x)/x) = 5.

Correct Answer: C — 5

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Q. Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

A. 1
B. 2
C. 3
D. 4

Solution

Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^3 + x^2 + x + 1) = 4.

Correct Answer: D — 4

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Q. Calculate the limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)

A. 0
B. 5/2
C. 2/5
D. ∞

Solution

Dividing the numerator and denominator by x^2, we get lim (x -> ∞) (5 + 3/x^2)/(2 + 1/x^2) = 5/2.

Correct Answer: B — 5/2

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Q. Determine the limit: lim (x -> 0) (tan(5x)/x) (2022)

A. 0
B. 1
C. 5
D. Undefined

Solution

Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 5. Thus, lim (x -> 0) (tan(5x)/x) = 5.

Correct Answer: C — 5

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Q. Determine the limit: lim (x -> 1) (x^3 - 1)/(x - 1) (2020)

A. 1
B. 2
C. 3
D. 0

Solution

Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^2 + x + 1. Evaluating at x = 1 gives 3.

Correct Answer: C — 3

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Q. Determine the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

A. 0
B. 1
C. 4
D. Undefined

Solution

Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Evaluating at x = 1 gives 4.

Correct Answer: D — Undefined

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Q. Evaluate the limit: lim (x -> 0) (tan(x)/x) (2023)

A. 0
B. 1
C. ∞
D. Undefined

Solution

Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.

Correct Answer: B — 1

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Q. Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)

A. 0
B. 1/6
C. 1/3
D. 1/2

Solution

Using the Taylor series expansion for sin(x), we find that lim (x -> 0) (x - sin(x))/x^3 = 1/6.

Correct Answer: B — 1/6

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Q. Evaluate the limit: lim (x -> 0) (x^3)/(sin(x)) (2022)

A. 0
B. 1
C. Infinity
D. Undefined

Solution

As x approaches 0, x^3 approaches 0 and sin(x) approaches 0, thus the limit is 0.

Correct Answer: A — 0

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Q. Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)

A. 3
B. 6
C. 9
D. Undefined

Solution

Factoring gives (x - 3)(x + 3)/(x - 3). Canceling (x - 3) gives lim (x -> 3) (x + 3) = 6.

Correct Answer: B — 6

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Q. Find the limit: lim (x -> 0) (x^2)/(sin(x)) (2023)

A. 0
B. 1
C. 2
D. Undefined

Solution

As x approaches 0, sin(x) approaches x, thus lim (x -> 0) (x^2/sin(x)) = lim (x -> 0) (x^2/x) = lim (x -> 0) x = 0.

Correct Answer: A — 0

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Q. Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)

A. 0
B. 1
C. 4
D. Undefined

Solution

Factoring gives ((x - 1)(x^3 + x^2 + x + 1))/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Thus, lim (x -> 1) = 4.

Correct Answer: A — 0

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Q. Find the limit: lim (x -> 2) (x^2 + 3x - 10)/(x - 2) (2021)

A. 0
B. 1
C. 5
D. 7

Solution

Factoring gives (x - 2)(x + 5)/(x - 2). For x ≠ 2, this simplifies to x + 5. Evaluating at x = 2 gives 7.

Correct Answer: D — 7

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Q. Find the limit: lim (x -> 2) (x^2 - 3x + 2)/(x - 2) (2021)

A. 1
B. 2
C. 0
D. Undefined

Solution

The expression is undefined at x=2. The limit does not exist as the function approaches infinity.

Correct Answer: D — Undefined

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Q. Find the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2023)

A. 0
B. 3
C. 6
D. 9

Solution

The expression can be factored as ((x - 3)(x + 3))/(x - 3). For x ≠ 3, this simplifies to x + 3. Thus, lim (x -> 3) (x + 3) = 6.

Correct Answer: A — 0

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Limits & Continuity MCQ & Objective Questions

Understanding "Limits & Continuity" is crucial for students preparing for various school and competitive exams in India. This topic forms the foundation of calculus and is frequently tested through MCQs and objective questions. Practicing these questions not only enhances conceptual clarity but also boosts confidence, helping students score better in their exams.

What You Will Practise Here

Definition and significance of limits in calculus
Types of limits: finite, infinite, and one-sided limits
Continuity and its types: pointwise and interval continuity
Key theorems related to limits and continuity
Application of limits in real-life scenarios
Common limit problems and their solutions
Graphical representation of continuous functions

Exam Relevance

The topic of Limits & Continuity is a staple in the syllabus for CBSE, State Boards, NEET, and JEE exams. Students can expect a variety of question patterns, including direct application of concepts, theoretical questions, and problem-solving scenarios. Familiarity with Limits & Continuity MCQ questions will greatly enhance your exam preparation and performance.

Common Mistakes Students Make

Confusing the concept of limits with continuity
Misinterpreting one-sided limits
Overlooking the importance of the epsilon-delta definition
Failing to apply theorems correctly in problem-solving

FAQs

Question: What are the basic properties of limits?
Answer: The basic properties include the limit of a sum, difference, product, and quotient of functions, provided the limits exist.

Question: How can I determine if a function is continuous?
Answer: A function is continuous if it is defined at a point, the limit exists at that point, and the limit equals the function's value.

Start solving practice MCQs on Limits & Continuity today to solidify your understanding and excel in your exams. Remember, consistent practice is the key to mastering this essential topic!

Limits & Continuity

Limits & Continuity MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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