Mathematics>Differential Calculus>Limits>Basics of Limits

HARD

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Let $L = \prod_{n = 3}^{\infty} (1 - \frac{4}{n^{2}}); M = \prod_{n = 2}^{\infty} (\frac{n^{3} - 1}{n^{3} + 1})$ and $N = \prod_{n = 1}^{\infty} \frac{{(1 + n^{-1})}^{2}}{1 + 2 n^{-1}}$ , then find the value of $L^{-1} + M^{-1} + N^{-1}$ .

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Important Questions on Limits

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to \frac{π}{2}} \frac{\cot x - \cos x}{{(π - 2 x)}^{3}}

equals

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

The value of

\underset{x \to 0}{l i m} \frac{x^{3} c o t x}{1 - c o s x}

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

For each positive real number

λ,

let

A_{λ}

be the set of all natural numbers

n

such that

| \sin (\sqrt{n + 1} - \sin n) | < λ .

Let

A_{λ}^{c}

be the complement of

A_{λ}

in the set of all natural numbers. Then-

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to 0} \frac{(\sqrt{1 + 2 x}) - 1}{x} =

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

If

[\cdot]

denote the greatest integer function then

\lim_{n \to \infty} \frac{[x] + [2 x] + \dots . + [n x]}{n^{2}}

is -

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to 0} (\frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x^{2}}) =

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

The limit

\underset{x \to \infty}{l i m} x^{2} \int_{0}^{x} e^{t^{3} - x^{3}} d t

equals

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

Let

f : R \to R

be a positive increasing function with

\lim_{x \to \infty} \frac{f (3 x)}{f (x)} = 1

. Then,

\lim_{x \to \infty} \frac{f (2 x)}{f (x)}

is equal to

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

The value of the

\lim_{x \to - \infty} (\sqrt{4 x^{2} - x} + 2 x)

is

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to \infty} (\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x})

is equal to

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to \infty} \frac{3 x^{3} + 2 x^{2} - 7 x + 9}{4 x^{3} + 9 x - 2}

is equal to

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

\underset{x \to \frac{π}{4}}{l i m} \frac{c o t^{3} x - t a n x}{c o s (x + \frac{π}{4})}

is

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

If

f (x) = \sqrt{\frac{x - \sin x}{x + \cos^{2} x}}

,

then

\lim_{x \to \infty} f (x)

is equal to

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to \infty} (\sqrt{x^{2} + 1} - \sqrt{x^{2} - 1})=

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

If

f

is differentiable at

x = 1

,

then

\lim_{x \to 1} \frac{x^{2} f (1) - f (x)}{x - 1}

is

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

Let

f : R \to R

be a function such that

\lim_{x \to \infty} f (x) = M > 0

. Then which of the following is false?

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

$\lim_{x \to 3} \frac{\sqrt{3 x} - 3}{\sqrt{2 x - 4} - \sqrt{2}}$ is equal to

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

$\lim_{x \to 0} \frac{{(27 + x)}^{\frac{1}{3}} - 3}{9 - {(27 + x)}^{\frac{2}{3}}}$ equals

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

$\lim_{x \to 1^{-}} \frac{\sqrt{π} - \sqrt{2 \sin^{- 1} x}}{\sqrt{1 - x}}$ is equal to

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

The value of $\underset{y \to 0}{l i m} \frac{\sqrt{1 + \sqrt{1 + y^{4}}} - \sqrt{2}}{y^{4}}$