Let f ( x ) = lim n → ∞ tan − 1 ( tan x ) 1 + ( log e x ) n , ∀ x ≠ ( 2 n + 1 ) π 2 , then

∀ x > e , f ( x ) is a constant function

∀ π 2 < x < π , the graph of f ( x ) is a staight line having y -intercept of − π

Let f ( x ) = lim n → ∞ tan − 1 ( tan x ) 1 + ( log e x ) n , ∀ x ≠ ( 2 n + 1 ) π 2 , then

∀ x > e , f ( x ) is a constant function

Let for all x > 0 , f ( x ) = lim n → ∞ n x 1 / n - 1 , then

f ( x y ) = x f ( x ) + y f ( y )

f ( x y ) = x f ( y ) + y f ( x )

HARD

Earn 100

Let $f (x) = \lim_{n \to \infty} \frac{\tan^{- 1} (\tan x)}{1 + {(\log_{e} x)}^{n}}, \forall x \neq (2 n + 1) \frac{π}{2}$ , then

14.29% studentsanswered this correctly

Important Questions on Limits

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

Let

\lim_{x \to 0} \frac{x 5^{x} - x}{1 - \cos x}

is equal to

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to - \infty} \frac{3 |x| - x}{|x| - 2 x} - \lim_{x \to 0} \frac{\log (1 + x^{3})}{\sin^{3} x} =

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

Let

0 < α < β < 1

. Then,

\lim_{n \to \infty} \int_{1 / (k + β)}^{1 / (k + α)} \frac{d x}{1 + x}

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

The value of

\lim_{x \to 0} \frac{(1 - \cos 2 x) (3 + \cos x)}{x \tan 4 x}

is equal to

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to 0} \frac{\sin ({πcos}^{2} x)}{x^{2}}

is equal to

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

The value of the limit

\lim_{θ \to 0} \frac{\tan (π \cos^{2} θ)}{\sin (2 π \sin^{2} θ)}

is equal to :

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{n \to \infty} (\frac{1}{3.7} + \frac{1}{7.11} + \frac{1}{11.15} + \dots + (n terms)) =

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to 0} \frac{x^{a} \sin^{b} x}{\sin x^{c}}, a, b, c, \in R ~ \{0\}

exists and has non-zero value, then

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to \infty} {(1 + \frac{a}{x} - \frac{4}{x^{2}})}^{2 x} = e^{3}

, then

a

is equal to

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

α

is the positive root of the equation,

p (x) = x^{2} - x - 2 = 0

, then

\lim_{x \to α^{+}} \frac{\sqrt{1 - \cos p (x)}}{x + α - 4}

is equal to

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

Let

f : R \to R

satisfy the equation

f (x + y) = f (x) \cdot f (y)

for all

x, y \in R

and

f (x) \neq 0

for any

x \in R

. If the function

f

is differentiable at

x = 0

and

f^{'} (0) = 3

, then

\lim_{h \to 0} \frac{1}{h} (f (h) - 1)

is equal to ___ .

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to 0} \frac{e^{x^{2}} - \cos x}{\sin^{2} x}

is equal to

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

Let

f (x) = \frac{1}{3} x \sin x - (1 - \cos x)

. The smallest positive integer

k

such that

\lim_{x \to 0} \frac{f (x)}{x^{k}} \neq 0

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

The value of

\lim_{h \to 0} \{\frac{\sqrt{3} \sin (\frac{π}{6} + h) - \cos (\frac{π}{6} + h)}{\sqrt{3} h (\sqrt{3} \cos h - \sin h)}\}

is :

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

Let for all

x > 0, f (x) = \lim_{n \to \infty} n (x^{1 / n} - 1),

then

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to 2} (\sum_{n = 1}^{9} \frac{x}{n (n + 1) x^{2} + 2 (2 n + 1) x + 4})

is equal to :

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{θ \to \frac{π}{4}} \frac{\sqrt{2} - \cos θ - \sin θ}{(4 θ - π)^{2}}

is equal to

HARD

Mathematics>Differential Calculus>Limits>Basics of Limits

f (x) = |\begin{array}{l} \sin x & \cos x & \tan x \\ x^{3} & x^{2} & x \\ 2 x & 1 & x \end{array}|,

x \in (- \frac{π}{2}, \frac{π}{2}),

then

\lim_{x \to 0} \frac{f (x)}{x^{2}}

is equal to

EASY

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to 0} \frac{1 - \cos m x}{1 - \cos n x} =

MEDIUM

Mathematics>Differential Calculus>Limits>Basics of Limits

\lim_{x \to 0} \frac{(1 - cos 2 x) (3 + cos x)}{x tan 4 x} =

Let f(x)=limn→∞tan−1(tanx)1+(logex)n, ∀ x≠(2n+1)π2, then

Important Questions on Limits

Learn and Prepare for any exam you want !

Let $f (x) = \lim_{n \to \infty} \frac{\tan^{- 1} (\tan x)}{1 + {(\log_{e} x)}^{n}}, \forall x \neq (2 n + 1) \frac{π}{2}$ , then