Evaluate Lim n → ∞ x + 2 x + 3 x + … … … . . + n x n 2 , where . denotes the greatest integer function.

We know x - 1 ≤ x < x , 2 x - 1 ≤ 2 x < 2 x , 3 x - 1 ≤ 3 x < 3 x , . . . . . . , n x - 1 ≤ n x < n x ⇒ x - 1 + 2 x - 1 + 3 x - 1 + . . . . + n x - 1 ≤ x + 2 x + 3 x + . . . . . . + n x < x + 2 x + 3 x + . . . . + n x ⇒ 1 + 2 + 3 + . . . + n x - n ≤ x + 2 x + 3 x + . . . . . . + n x < 1 + 2 + 3 + . . . + n x ⇒ n n + 1 2 x - n ≤ x + 2 x + 3 x + . . . . . . + n x < n n + 1 2 x ⇒ n n + 1 2 x - n n 2 ≤ x + 2 x + 3 x + . . . . . . + n x n 2 < n n + 1 2 x n 2 ⇒ lim n → ∞ 1 + 1 n 2 x - 1 n ≤ lim n → ∞ x + 2 x + 3 x + . . . . . . + n x n 2 < lim n → ∞ 1 + 1 n 2 x ⇒ 1 2 x - 0 ≤ lim n → ∞ x + 2 x + 3 x + . . . . . . + n x n 2 < 1 2 x ⇒ lim n → ∞ x + 2 x + 3 x + . . . . . . + n x n 2 = x 2

Evaluate Lim n → ∞ x + 2 x + 3 x + … … … . . + n x n 2 , where . denotes the greatest integer function.

We know x - 1 ≤ x < x , 2 x - 1 ≤ 2 x < 2 x , 3 x - 1 ≤ 3 x < 3 x , . . . . . . , n x - 1 ≤ n x < n x ⇒ x - 1 + 2 x - 1 + 3 x - 1 + . . . . + n x - 1 ≤ x + 2 x + 3 x + . . . . . . + n x < x + 2 x + 3 x + . . . . + n x ⇒ 1 + 2 + 3 + . . . + n x - n ≤ x + 2 x + 3 x + . . . . . . + n x < 1 + 2 + 3 + . . . + n x ⇒ n n + 1 2 x - n ≤ x + 2 x + 3 x + . . . . . . + n x < n n + 1 2 x ⇒ n n + 1 2 x - n n 2 ≤ x + 2 x + 3 x + . . . . . . + n x n 2 < n n + 1 2 x n 2 ⇒ lim n → ∞ 1 + 1 n 2 x - 1 n ≤ lim n → ∞ x + 2 x + 3 x + . . . . . . + n x n 2 < lim n → ∞ 1 + 1 n 2 x ⇒ 1 2 x - 0 ≤ lim n → ∞ x + 2 x + 3 x + . . . . . . + n x n 2 < 1 2 x ⇒ lim n → ∞ x + 2 x + 3 x + . . . . . . + n x n 2 = x 2

Let f : R → R be a function such that lim x → ∞ ⁡ f x = M > 0 . Then which of the following is false ?

lim x → ∞ ⁡ x sin ⁡ e - x f x = M

lim x → ∞ ⁡ x sin ⁡ 1 x f x = M

lim x → ∞ ⁡ sin ⁡ f x = sin ⁡ M

lim x → ∞ ⁡ sin ⁡ x x ⋅ f x = 0

For each positive real number λ , let A λ be the set of all natural numbers n such that | sin ⁡ ( n + 1 - sin ⁡ n ) | < λ . Let A λ c be the complement of A λ in the set of all natural numbers. Then-

A 1 2 c , A 1 3 c , A 2 5 c are all finite sets

A 1 2 , A 1 3 , A 2 5 are all finite sets

A 1 3 is a finite set but A 1 2 , A 2 5 are infinite sets

A 1 3 , A 2 5 are finite sets and A 1 2 is an infinite set

MEDIUM

Earn 100

Evaluate $\underset{n \to \infty}{Lim} \frac{[x] + [2 x] + [3 x] + \dots \dots \dots . . + [n x]}{n^{2}}$ , where $[.]$ denotes the greatest integer function.

Important Questions on Limits

MEDIUM

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

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

MEDIUM

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

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

equals

EASY

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

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

HARD

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

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

EASY

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

\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})}

MEDIUM

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

f

is differentiable at

x = 1

,

then

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

EASY

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

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

EASY

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

For each

t \in R,

let

[t]

be the greatest integer less than or equal to

t

. Then

\lim_{x \to 0^{+}} x ([\frac{1}{x}] + [\frac{2}{x}] + \dots + [\frac{15}{x}])

EASY

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

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

equals

HARD

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

The limit

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

equals

MEDIUM

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

Let

a > 0

be a real number. Then the limit

\lim_{x \to 2} \frac{a^{x} + a^{3 - x} - (a^{2} + a)}{a^{3 - x} - a^{\frac{x}{2}}}

is-

HARD

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

[\cdot]

denote the greatest integer function then

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

is -

HARD

Mathematics>Differential Calculus>Limits>Limits of Trigonometric Functions

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?

EASY