Differentiability Questions - JEE

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:

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

If a is the positive root of the equation p(x) = x2 − x − 2 = 0, then limx → a 1 − cos(πp(x))/x2 + a − 4 is equal to

The value of $\lim_{x \to \pi/4} (1 + [x])^{\log(\tan x)}$ (where $[\cdot]$ denotes greatest integer function) is

If $\lim_{x \to 0} \frac{1}{x^8}\left[\cos\frac{x^2}{2} - \cos\frac{x^2}{4} + \cos\frac{x^2}{2} - \cos\frac{x^2}{4}\right] = 2k$, then the value of $k$ is ……… .

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

If f(x) = cos x × cos 2x × cos 4x × cos 8x × cos 16x, then f'(π/4) is

If y = sin⁻¹((1-x)/(1+x)), 0 , then dy/dx is

The points of discontinuity of $y = \frac{1}{u^2 + u - 2}$, where $u = \frac{1}{x-1}$, are:

Given $f(0) = 0$ and $f(x) = \frac{1}{1-e^{-1/x}}$ for $x \neq 0$. Then, only one of the following statements on $f(x)$ is true.

If x\log_e(\log_e x) - x^2 + y^2 = 4 (y > 0), then \frac{dy}{dx} at x = e is equal to

If $f(x) = \cos\left[\frac{\pi}{x}\right]\cos\left(\frac{\pi(x-1)}{2}\right)$, where $[\cdot]$ denotes the greatest integer function, then $f(x)$ is continuous at

Let $(\tan\alpha)x + (\sin\alpha)y = \alpha$ and $(\alpha\operatorname{cosec}\alpha)x + \cos\alpha y = 1$ be two variable straight lines, $\alpha$ being the parameter. Let P be the point of intersection of the lines. If the coordinates of P in the limiting position when $\alpha \to 0$ be $(h, k)$ then

Given,\[f(x) = \begin{cases} \frac{\cos 2x - \sin 2x - 1}{x^2 + x - 2}, & x \neq \text{?} \\ a, & x = \text{?} \end{cases}\]where f is continuous at the point where the function is defined piecewise. Find the constant a.

If f'(x) = g(x) and g'(x) = -f(x) for all x and f(2) = 4 = f'(2), then (f(24))^2 + (g(24))^2 is

If the function f(x) = {a|p - x| + 1, x ≤ 5; b|x - p| + 3, x > 5} is continuous at x = 5, then the value of a - b is

If $y = \tan^{-1}(\sec x - \tan x)$, then $\frac{dy}{dx}$ is equal to

If $\displaystyle \lim_{x \to a} \frac{f(x) - f(a)}{(x-a)^3}$ is a finite non-zero number, then $f(x)$ is of maximum degree

The value of $\lim_{x \to 0} \sin^{-1}\{x\}$ (where $\{\cdot\}$ denotes fractional part of $x$) is

$\lim_{x \to \infty} C_n^m x^m \left(1 + \frac{1}{n}\right)^{nx} \left(1 + \frac{1}{x}\right)^m$ equals to

Let $h(x) = \min\{x; x^2\}$ for every real number $x$. Then, which one of the following is true?

The value of $\lim_{x \to 0} \frac{\tan(\{1/x\}) \sin(ax)}{\sin(bx)}$, where $\{x\}$ denotes the fractional part function is

If f(x) = \begin{cases} A \sin[x] & \text{for } [x] \geq 0 \\ 0 & \text{for } [x] , where $[x]$ denotes the greatest integer less than or equal to $x$, then $\lim_{x \to 0} f(x)$ equals

Differentiability Questions (1063)