Limits Questions - JEE

If $y^2 = P(x)$ which is a polynomial of degree 3, then $2\dfrac{d}{dx}\left(y^3 \dfrac{d^2y}{dx^2}\right)$ equals

If limx→0 eax−cos(bx)−cxe−cx 2 1−cos 2x = 17, find 5a2 + b2.

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

Let f(x) = ( sin x x ∈Z 0 x /∈Z and g(x) =      x2 + 1 x ̸= 0, 2 4 x = 0 5 x = 2 , then:

limx→1− √π− √ 2 sin−1 x √1−x is equal to:

Given $(2x)^{2y} = 4e^{2x-2y}$, then $\dfrac{dy}{dx}(1 + \log_e 2x)^2$ equals:

Limit check for piecewise f(x):

Let $ S = \{t \in R : f(x) = |x - \pi| \cdot (e^{|x|} - 1)\sin|x| $ is not differentiable at $t\}$. Then the set S is equal to:

If limx→1 x+x2+···+xn−n x−1 = 820, then n is equal to:

Find g(x) = −x4b where b = limx→∞( √ x2 + x + 1 − √ x2 + 1).

If limx→1 x2−ax+b x−1 = 5, then a + b is equal to:

Find $\displaystyle\lim_{x \to \pi/2} \frac{\left[1 - \tan\left(\dfrac{x}{2}\right)\right]\left[1 - \sin x\right]}{\left[1 + \tan\left(\dfrac{x}{2}\right)\right]\left[\pi - 2x\right]^3}$.

Number of solutions of f(x) + g(x) = 0?

limx→2 3x+33−x−12 3−x/2−31−x is equal to:

limx→0 sin2(π cos4 x) x4 is equal to:

If $f(0) = 1$ and $\displaystyle\lim_{t \to x} \frac{\sec x \cdot f(t) - f(x) \sec t}{t - 1} = \sec^2 x$. The value of $\dfrac{f(0)}{f'(0)}$, is:

The value of $\displaystyle\lim_{x \to 0^+} \dfrac{\displaystyle\int_0^{\arctan x} (\sin t^2)\, dt}{x \cos x - x}$ is equal to:

$\lim_{x \to 1^-} \dfrac{\sqrt{\pi} - \sqrt{2\sin^{-1}x}}{\sqrt{1-x}}$ is equal to __________ (up to four decimal places).

Let [x] denote the greatest integer function. Find limx→0 tan(π sin2 x)+(|x|−sin |x|)[x]2 x2 :

If $ f $ is a real-valued differentiable function satisfying $ |f(x) - f(y)| \leq (x-y)^2 $, $ x, y \in R $ and $ f(0) = 0 $, then $ f(1) $ equals

Let $f: R \to R$ defined by $f(x) = x^3 + 3x + 1$ and $g$ be the inverse of $f$, then the value of $g''(5)$ equals:

$\lim_{h \to 0} \frac{2\left[\sqrt{3}\sin\left(\frac{\pi}{6}+h\right) - \cos\left(\frac{\pi}{6}+h\right)\right]}{\sqrt{3}h(\sqrt{3}\cos h - \sin h)}$ is equal to

If $f(x) = \frac{\sin[x]}{[x]}, [x] \neq 0$$= 0, [x] = 0$where $[x]$ is the greatest integer function, then $\lim_{x \to 0} f(x)$ equals:

If $f(x) = \int_0^x |t| dt$, then

limx→π/2 tan2 x[ p 2 sin2 x + 3 sin x + 4 − p sin2 x + 6 sin x + 2]

If f(x) is an odd linear polynomial with f(1) = 1, then limx→0 2f(tan x)−2f(sin x) x2f(sin x) is:

Let f(x) = tan x x , then the value of limx→0([f(x)] + x2)1/{f(x)} is:

limx→1+ (1−|x|+sin |1−x|) sin( π 2 [1−x]) |1−x|[1−x] is equal to: (1) −1 (2) 1 (3) D.N.E (4) 0

$ f(x) = [\log_e x] + \sqrt{\{\log_e x\}}, x > 1 $, where [.] and {.} denote the greatest integer function and the fractional part function respectively, then

Properties of f(x) = limn→∞ (tan x)2n+x2 sin2 x+(tan x)2n for x ∈(−π/2, π/2):

Assume that $f$ is continuous on $[a, b]$, $a > 0$ and differentiable on $(a, b)$. If $\dfrac{f(a)}{a} = \dfrac{f(b)}{b}$, then there exists $x_0 \in (a, b)$ such that:

limn→∞ 1+2−3+···+(3n−2)+(3n−1)−3n √ 2n4+4n+3− √ n4+5n+4

If f(x) = cos 2−cos 2x x2−|x| , then:

For each x ∈R, let [x] be the GIF. Then limx→0−x([x]+|x|) sin[x] |x| is equal to: (1) −sin 1 (2) 0 (3) 1 (4) sin 1

limn→∞ en (1+ 1 n) n2 equals:

The value of the limit ℓis:

Let $f(x) = \frac{\log_e(1+ax) - \log_e(1-bx)}{x}, x \neq 0$. The value to be assigned to $f(x)$ at $x = 0$ so that $f(x)$ is continuous at $x = 0$ is:

If $f(x) = \log_e x$ then the differential coefficient of $f(\log_e x)$ with respect to $x$ is

Let $a_n = \lim_{n \to 0}\left(\dfrac{a_{n-1}}{n}\right)^2 (f(h) - f(0))^2$ where $f'(0) = 1$. If $a_1 = 1$, find $\prod_{i=1}^{10} a_i$.

If $\lim_{x \to s} f(x)$ and $\lim_{x \to s} g(x)$ exist then $\lim_{x \to s} g(x)$ exist.

Let Un = n! (n+2)! where n ∈N. If Sn = Pn r=1 Ur, then limn→∞Sn equals:

The value of d/dx tan2(x/(1-tan2x))cot 3x is

The value of $\lim_{x \to 0}(\sin x)^x$ is

If f(x) = 1 + cos2(x2), then the value of f′(π/6) is

If y = sin x + y, then dy/dx is equal to

The value of limx→∞ √ x2 −2x −1 − √ x2 −7x + 3 is:

Let L = limx→0 a− √ a2−x2−x2 4 x4 , a > 0. If L is finite, then:

$$\lim_{n \to \infty} \frac{n^2}{((n^2 + 1^2)(n^2 + 2^2) \dots (n^2 + n^2))^\frac{1}{n}}$$ equals:

Find the value of $\lim_{x \to 0} \frac{\sin^2 x}{2 - \sqrt{1 + \cos x}}$

If $\lim_{x \to 0} \left[1 + x + \frac{f(x)}{x}\right]^{1/x} = e^3$, then $\lim_{x \to 0} \frac{f(x)}{x}$ is equal to

Limits Questions (1092)