Mathematics

Limits and Continuity

49 Questions

Limits and continuity explore the foundational behavior of mathematical functions as they approach specific values or infinity. The syllabus covers evaluating sequence limits, trigonometric boundaries, and logarithmic expressions. These rigorous quantitative aptitude questions are standard in collegiate and engineering assessments.

Sequence limit evaluationInfinity limit functionsTrigonometric limit valuesLogarithmic limitsContinuity theorems

Limits and Continuity Questions

Multiple choice taylor's and maclaurin's series applications of differential calculus maths

The value of $\mathop {\lim }\limits _{x \to 0} \frac{{\sin x + \log \left( {1 - x} \right)}}{{{x^2}}}$ equals

  1. $0$
  2. $\frac{1}{2}$
  3. $\frac{{ - 1}}{2}$
  4. $-1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\mathop {\lim }\limits _{x \to 0} {{\sin x + \log \left( {1 - x} \right)} \over {{x^2}}}$

${0 \over 0}form$

$ = \mathop {\lim }\limits _{x \to 0} {{\cos x - {1 \over {1 - x}}} \over {2x}}$

(L-hospital)

$ = \mathop {\lim }\limits _{x \to 0} {{\left( {1 - x} \right)\cos x - 1} \over {2x\left( {1 - x} \right)}}$

$ = \mathop {\lim }\limits _{x \to 0} {{\cos x - x\cos x - 1} \over {2x\left( {1 - x} \right)}}$

${0 \over 0}form$

$ = \mathop {\lim }\limits _{x \to 0} {{ - \sin x + x\sin x - \cos x} \over { - 2x + 2\left( {1 - x} \right)}}$

$ = {{0 + 0 - 1} \over {0 + 2}}$

$ =  - {1 \over 2}$

Multiple choice maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

$\lim _{ x\leftarrow 1 }{ \cfrac { x+{ x }^{ 2 }+{ x }^{ 3 }+....+{ x }^{ n }-n }{ x-1 }  } =$

  1. $\cfrac{n(n+1)}{2}$
  2. $\cfrac{n+1}{2}$
  3. $\cfrac{2}{n}$
  4. $n$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\underset{x \rightarrow 1}{lim} \dfrac{x + x^2 + x^3  + .... + x^n - x}{x - 1} \left(\dfrac{0}{0} \right)$ form

By L Hospital rule
$= \underset{x \rightarrow 1}{lim} \dfrac{1 + 2x + 3x^2 + ... + nx^{n - 1}}{1}$
$= 1 + 2 + 3 + ... + n$
$= \dfrac{n (n + 1)}{2}$

Multiple choice

What does the symbol (\lim) represent?

  1. Limit

  2. Derivative

  3. Integral

  4. Exponential
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(\lim) is a mathematical symbol that represents the limit of a function. It is used to find the value that a function approaches as the input approaches a certain value.

Multiple choice

What is the limit point of the set {1/n : n is a natural number} in the real numbers?

  1. 0

  2. 1

  3. ∞

  4. -∞
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The limit point of a set is a point that is arbitrarily close to infinitely many points in the set. In this case, the set {1/n : n is a natural number} has a limit point of 0 because for any ε > 0, we can find a natural number N such that 1/N < ε, which means that there is a point in the set that is less than ε away from 0.

Multiple choice

Which theorem states that the limit of a sequence ((a_n)) is (L) if and only if for every (\epsilon > 0), there exists a positive integer (N) such that (|a_n - L| < \epsilon) whenever (n > N)?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Angle Sum Theorem

  4. Limit Theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Limit Theorem states that the limit of a sequence ((a_n)) is (L) if and only if for every (\epsilon > 0), there exists a positive integer (N) such that (|a_n - L| < \epsilon) whenever (n > N).

Multiple choice

Which theorem states that the limit of a sequence ((a_n)) is (L) if and only if for every (\epsilon > 0), there exists a positive integer (N) such that (|a_n - L| < \epsilon) whenever (n > N)?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Angle Sum Theorem

  4. Limit Theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Limit Theorem states that the limit of a sequence ((a_n)) is (L) if and only if for every (\epsilon > 0), there exists a positive integer (N) such that (|a_n - L| < \epsilon) whenever (n > N).

Multiple choice

Which theorem states that the limit of a sequence ((a_n)) is (L) if and only if for every (\epsilon > 0), there exists a positive integer (N) such that (|a_n - L| < \epsilon) whenever (n > N)?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Angle Sum Theorem

  4. Limit Theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Limit Theorem states that the limit of a sequence ((a_n)) is (L) if and only if for every (\epsilon > 0), there exists a positive integer (N) such that (|a_n - L| < \epsilon) whenever (n > N).

Multiple choice

Let $f(x) = \frac{1}{x}$. Find the limit of $f(x)$ as $x$ approaches 0.

  1. 0

  2. 1

  3. Does not exist

  4. Infinity
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The limit of $f(x)$ as $x$ approaches 0 does not exist because the function approaches both positive and negative infinity as $x$ approaches 0 from the positive and negative sides, respectively.

Multiple choice

Find the value of (\lim_{x \to 0} \frac{\sin(3x)}{x}).

  1. 0

  2. 1

  3. 3

  4. Does not exist
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find the limit, we can use L'Hopital's rule: (\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{\frac{d}{dx}[\sin(3x)]}{\frac{d}{dx}[x]} = \lim_{x \to 0} \frac{3\cos(3x)}{1} = 3\cos(0) = 3).

Multiple choice

The equation (\lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n = e) is known as:

  1. Euler's number

  2. Fermat's Last Theorem

  3. Ramanujan's conjecture

  4. Stirling's approximation
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation (\lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n = e) is known as Euler's number, which is one of the most important constants in mathematics.

Multiple choice

Let (f(x) = \frac{x^2 - 1}{x - 1}). Find the value of (\lim_{x \to 1} f(x)).

  1. 0

  2. 1

  3. 2

  4. Does not exist
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We can factor the numerator of (f(x)) as (x^2 - 1 = (x + 1)(x - 1)). Therefore, (f(x) = \frac{(x + 1)(x - 1)}{x - 1} = x + 1). Substituting (x = 1) into (f(x)), we get (f(1) = 1 + 1 = 2). Therefore, (\lim_{x \to 1} f(x) = 2).