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

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

MEDIUM

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Suppose $\lim_{x \to 0} \frac{f (x)}{x^{3}}$ exists and is equal to $L$ , where $f (x) = |\begin{matrix} a + \sin \frac{x}{2} & - bcos x & 0 \\ - bcos x & 0 & a + \sin \frac{x}{2} \\ 0 & a + \sin \frac{x}{2} & - bcos x \end{matrix}|$ Then, $- 112 L$ is equal to _____ .

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Important Questions on Limits

EASY

Mathematics>Differential Calculus>Limits>Methods of Evaluation of Limits

What is

\lim_{x \to 0} \frac{e^{x} - (1 + x)}{x^{2}}

equal to?

EASY

Mathematics>Differential Calculus>Limits>Methods of Evaluation of Limits

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

MEDIUM

Mathematics>Differential Calculus>Limits>Methods of Evaluation of Limits

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

HARD

Mathematics>Differential Calculus>Limits>Methods of Evaluation 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

EASY

Mathematics>Differential Calculus>Limits>Methods of Evaluation 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})}

MEDIUM

Mathematics>Differential Calculus>Limits>Methods of Evaluation of Limits

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>Methods of Evaluation of Limits

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

EASY

Mathematics>Differential Calculus>Limits>Methods of Evaluation of Limits

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}])

HARD

Mathematics>Differential Calculus>Limits>Methods of Evaluation 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>Methods of Evaluation 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>Methods of Evaluation of Limits

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-

MEDIUM

Mathematics>Differential Calculus>Limits>Methods of Evaluation of Limits

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

is equal to

HARD

Mathematics>Differential Calculus>Limits>Methods of Evaluation 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?

EASY