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

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If $x$ is a real number in $[0, 1]$ then the value of $f (x) = \lim_{m \to \infty} \lim_{n \to \infty} \{1 + \cos^{2 m} (n! π x)\}$ is given by:

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Important Questions on Limits

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})}

is

MEDIUM

Mathematics>Differential Calculus>Limits>Methods of Evaluation 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>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

If

f (x)

is continuous and

f (\frac{9}{2}) = \frac{2}{9},

then

\lim_{x \to 0} f (\frac{1 - \cos 3 x}{x^{2}})

equals to

EASY

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

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

equals

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?

MEDIUM

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

$\underset{x \to 0}{l i m} \frac{{s i n}^{2} x}{\sqrt{2} - \sqrt{1 + c o s x}}$ 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

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

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 value of the $\lim_{x \to - \infty} (\sqrt{4 x^{2} - x} + 2 x)$ is

HARD

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

$\lim_{x \to 0} \frac{{(1 - \cos 2 x)}^{2}}{2 x \tan x - x \tan 2 x}$ is

MEDIUM

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

The value of the $\lim_{x \to 0} {(\frac{x}{\sin x})}^{6 / x^{2}}$ is-

HARD

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

$\underset{x \to 2}{l i m} \frac{3^{x} + 3^{3 - x} - 12}{3^{- \frac{x}{2}} - 3^{1 - x}}$ is equal to

EASY

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

If $\lim_{x \to 1} \frac{x^{2} - a x + b}{x - 1} = 5,$ then $a + b$ is equal to:

EASY

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

If $\underset{x \to 1}{l i m} \frac{x^{4} - 1}{x - 1} = \underset{x \to k}{l i m} \frac{x^{3} - k^{3}}{x^{2} - k^{2}}$ , then $k$ is

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

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

If $[\cdot]$ denote the greatest integer function then $\lim_{n \to \infty} \frac{[x] + [2 x] + \dots . + [n x]}{n^{2}}$ is -