Continuity Questions - JEE

For any positive integer n, define fn(x) = n X j=1 tan−1 1 1 + (x + j)(x + j −1) , x ∈(0, ∞). Then which of the following statement(s) is(are) true?

The value of limx→−1 cos 2−cos 2x x2−|x| is:

252. The number of points where $f(x) = |x + [x]| - 3[2x] + 4[3x]$ is discontinuous in $[-1, 1]$, is:[Note: $[k]$ denotes greatest integer less than or equal to $k$.]

If β = limx→0 ex3−(1−x3)1/3+((1−x2)1/2−1) sin x x sin2 x , find 6β.

Let $y(x)=\left(1+x\right)\!\left(1+x^2\right)\!\left(1+x^4\right)\!\left(1+x^8\right)\!\left(1+x^{16}\right)$. Then $y'(1)-y(1)$ is:

If $3f(x) - 2f(1/x) = x$, then $f'(2)$ is equal to:

For n ∈N, let an = Pn k=1 2k and bn = Pn k=1(2k −1). Then limn→∞(√an −√bn) is:

Let f: \mathbb{R}^+ \to \mathbb{R} be a differentiable function satisfying: \frac{f(x)}{y} + \frac{f(y)}{x} + f(xy) = \frac{}{} for all x, y \in \mathbb{R}^+, with f(1) = 0 and f'(1) = 1. Find \lim_{x \to e} \lfloor f(x) \rfloor (where \lfloor \cdot \rfloor denotes greatest integer function).

Let α(a) and β(a) be the roots of the equation ( 3√1 + a−1)x2+(√1 + a−1)x+( 6√1 + a−1) = 0. Find lima→0+ α(a) and lima→0+ β(a).

If x = \cos\theta and y = \sin^3\theta, then find y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 at \theta = \frac{\pi}{2}.

Let $f$ be a continuous function on $\mathbb{R}$ such that $f\left(\frac{1}{n^4}\right) = \frac{(\sin e^{-n^2})e^{-n^2} + \frac{n^2}{n^2+1}}{1}$. Then $f(0) = $

If $f(x) = \cot^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)$ and $g(x) = \cos^{-1}\left(\frac{1 - x^2}{1 + x^2}\right)$ then $\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}$ for \(0

Let $S$ be the set of all points in $(-\pi,\pi)$ at which the function $f(x)=\min(\sin x,\cos x)$ is not differentiable. Then $S$ is a subset of which of the following?

Let $y=y(x)$ be a function of $x$ satisfying $y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$ where $k$ is a constant and $y\!\left(\tfrac{1}{2}\right)=-\tfrac{1}{4}$. Then $\dfrac{dy}{dx}$ at $x=\tfrac{1}{2}$:

Let $f$ be defined for all $x \in \mathbb{R}$. If $f$ is differentiable and $f(x^3) = x^5$ for all $x \in \mathbb{R}\,(x\ne0)$, then $f'(27)$ is equal to:

If $y = f(x)$ satisfies $f(x+y) = f(x)+f(y)+2xy-1$ for all $x,y\in\mathbb{R}$ and $f'(0) = \cos\alpha$, then which of the following is/are true?

$h(x)=\min\{\{x\},1-\{x\}\}$. h is:

Let $f$, $g$ and $h$ be differentiable functions. If $f(0) = 1$, $g(0) = 2$, $h(0) = 3$ and the derivative of their pairwise product at $x = 0$ are $(fg)'(0) = 6$, $(gh)'(0) = 4$ and $(hf)'(0) = 5$, then compute the value of $(fgh)'(0)$.

771. Let $f: R \to R$ be a function defined by $f(x) = \begin{cases} x^3 + 2x^2 + x + c, & \text{if } x \leq b \\ e^x, & \text{if } x > b \end{cases}$, where $b$ and $c$ are integers. If $f(x)$ is differentiable $\forall\, x \in R$, then find the value of $(b + c)$.

Let $f(x) = \begin{cases} \dfrac{ax^3 + bx^2 + cx + d}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}$ where $a, b, c, d$ are in A.P. If $f(x)$ is continuous, then the number of local minima of $y = |f(|x|)|$ is:

Let \(g(x) = \begin{cases} k\sqrt{x+1}; & 0 \leq x \leq 3 \\ mx + 2; & 3

A certain function $f(x)$ has the property that $f(3x) = af(x)$ for all positive real values of $x$ and $f(x) = 1 - |x - 2|$ for $1 \le x \le 3$. Find $\lim_{x \to 2} (f(x))^{\frac{\csc\left(\frac{\pi x}{2}\right)}{1}}$

If $f(x) = \begin{cases} x\tan^{-1}(1/x), & x \in (-\infty,-1]\cup[1,\infty) \\ 0, & x = 0 \end{cases}$, then $f'(0)$ is:

If \(\alpha,\ \beta,\ (\alpha

Question 23: Given We have f: (-1, 1) → ℝf(0) = -1, f'(0) = 1If g(x) = [f(2f(x) + 2)]^(1/2)Find g'(0)

The value of limx→0(1 + sin x)cos x is:

Evaluate: $\lim_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{\sin x}{x - \sin x}}$

If $f(x) = \sin\!\left[\tan^{-1}\!\left(\dfrac{1-x^2/2}{x}\right)\right]$, $g(x) = \cos\!\left[2\tan^{-1}\!\left(\dfrac{x}{1+\sqrt{1-x^2}}\right)\right]$, and $h(x)=f(x)-g(x)$, then $h'(1/2)$:

\(f(x)=\begin{cases}|x-3| & x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4} & x is:

limx→0 cos(sin x)−cos x x4 is equal to:

Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in A.P., then $f'(a),\, f'(b),\, f'(c)$ are in:

If $f(\theta) = \tan^{-1}\!\left(\dfrac{\sin\theta-\cos\theta}{\sin\theta+\cos\theta}\right)$, then which of the following are correct?

If limx→0 x2 sin βx αx−sin x = 1, find 6(α + β).

Let $y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $\dfrac{d}{dx}\!\big[y(x)\big]_{x=1}$ equals: [Integer type]

Let $|f(x)-f(y)|\le 2|x-y|^{3/2}$, $f(0)=1$. Find $\int_0^1 f^2(x)\,dx$.

Given $f$ piecewise defined. $f_1(x)=|f(|x|)|$. If $f$ is continuous in $[-2,10]$, then:

The value of limx→0 | cos(sin 3x)|−1 x2 is:

Evaluate: $\lim_{x \to 0} \left(\tan \frac{x}{x}\right)^{\frac{1}{x}}$

Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in A.P., then $f'(a)$, $f'(b)$, $f'(c)$ are in:

Suppose $f$ is a derivable function that satisfies the equation $ f(x+y) = f(x) + f(y) + x^2y + xy^2 $ for all real numbers $x$ and $y$. Suppose that $ \lim_{x \to 0}\left[\dfrac{f(x)}{x}\right] = 1 $, find $ f(3) = $ __________.

The value of $\lim_{x \to 0} \frac{(\tan x - \sin x)}{(\tan x - \sin x) + (\tan x + \sin x)} + \frac{1}{r^3 - r}$ is

882. Let $f(x) = \begin{cases} \dfrac{ax^3 + bx^2 + cx + d}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}$ be a continuous function where $a, b, c, d$ are in arithmetic progression. Then find the number of points where $|f(|x|)|$ is non derivable.

The value of $\lim_{n \to \infty} \sqrt[n]{a^n + b^n}$ (where $a > 0, b > 0$) is equal to

If $y = e^{\sin^{-1}x} + e^{\cos^{-1}x}$, then which of the following is/are true?

Let fn(x) = limt→x $f_n(x) = \lim_{t \to x} \frac{\sin^{-1}(nt)}{2t} = \frac{\sin^{-1}(nx)}{2x}$. Find the value of $\lim_{x \to 0} \left[\frac{\sin^2 2x}{2x}\right] + \left[\frac{\sin^{-1} 4x}{2x}\right]$.

279. Let $ f(x) = \dfrac{x \ln x - \ln x}{9x^2 - 2e^x x - 9x + 2e^x} + 2 $ and $ g(x) = \sin^2\left(\dfrac{\pi x^2}{2}\right) $, then the value of $ \lim_{x \to 1} \dfrac{f(x)}{g(x)} $ is:

For the curve \sin x + \sin y = 1 lying in the first quadrant, there exists a constant a for which \lim_{x \to 0} x^a \frac{d^2y}{dx^2} = L (not zero). Find 2a.

Evaluate: $\lim_{x \to \infty} \left(\frac{3x^2 + 1}{4x^2 - 1}\right)^{\frac{x^2}{1+x}}$

Let $ f(x) = \cot^{-1}\left(\text{sgn}\left(\dfrac{[x]}{2x - [x]}\right)\right) $:Statement-1: $ f(x) $ is discontinuous at $ x = 1 $.Statement-2: $ f(x) $ is non-differentiable at $ x = 1 $.Which of the following option is correct?[Note: $[k]$ denotes greatest integer function less than or equal to $k$.]

Let a = min{x2 + 2x + 3, x ∈R} and b = limθ→0 1−cos θ θ2 . Then the value of Pn r=0 arbn−r is:

Continuity Questions (1086)