Continuity Questions - JEE

In [1,3], the function $ [x^2 + 1][.] $ denoting the greatest integer function, is continuous -

If f(x) = $\frac{1}{(x-1)(x-2)}$ and g(x) = $\frac{1}{x^2}$, then set of points in domain of fog(x) at which fog(x) is discontinuous.

The function f(x) = [x]. cos $\frac{2x-1}{2\pi}$, where [·] denotes the greatest integer function, is discontinuous at :-

A function f(x) is defined as f(x) = $\frac{A \sin x + \sin 2x}{x^3}$, (x ≠ 0). If the function is continuous at x = 0, then -

The function f(x) = $ [x^2 - [x]^{2}] $ (where [y] is the greatest integer less than or equal to y), is discontinuous at :

For the function f(x) = $ \begin{cases} \frac{1}{x + 2 - \sqrt{x}} & \text{if } x \neq 2 \\ k & \text{if } x = 2 \end{cases} $ which of the following holds?

Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min\{\sqrt{2}x,\,x^2\}$. Let $S=\{x\in(-2,2): g(x)=|x|\,[x^2]$ is discontinuous at $x\}$. Then $\displaystyle\sum_{x\in S}f(x)$ equals

If $f(x) = \begin{cases} \dfrac{a|x|+x^2-2\sin|x|\cos|x|}{x} & x\neq0 \\ b & x=0 \end{cases}$ is continuous at $x=0$, then $a+b$ is equal to

If $f(x) = \begin{cases} \frac{[(a-n)nx - \tan x]\sin nx}{x^2} & \text{at } x = 0 \\ 0 & \text{at } x = 0 \end{cases}$, where $n$ is a non-zero real number, and $f$ is continuous at $x = 0$, then $a$ is equal to:

Let $f$ be a function defined on $(-\pi/2, \pi/2)$ as follows: $f(x) = \begin{cases} \frac{2^{[1/n]} - [x] - \frac{[x]}{[n2-1]}}{x\tan x} & x \neq 0 \\ k & x = 0 \end{cases}$. The value of $k$ so that $f$ is continuous at $x = 0$ is:

If $f(x)=x(\sqrt{x}-\sqrt{x+1})$, then:

If $f(x)=\text{sgn}(x^5)$, which are false?

Let $S$ be the set of all points where $\sqrt[5]{x^2|x|^3}-\sqrt[3]{x^2|x|}-1$ is not differentiable. $S$ is a subset of:

Number of points where $f(x)=|x-\text{sgn}(x)|$ is non-differentiable (sgn(.) denotes signum function):

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function $f(x) = \begin{cases} b^2\sin\!\left(\dfrac{\pi}{2}\!\left[\dfrac{\pi}{2}(\cos x+\sin x)\cos x\right]\right) & x < 0 \\[6pt] \dfrac{\sin x - \tfrac{1}{2}\sin 2x}{x^3} & x > 0 \\[4pt] a & x = 0 \end{cases}$ is continuous at $x=0$, then $a^2+b^2$ is equal to

Let $f(x)=\begin{cases}x^3+2x^2 & x\in\mathbb{Q}\\ -x^3+2x^2+ax & x\notin\mathbb{Q}\end{cases}$. Integral value of $a$ so that $f$ is differentiable at $x=1$:

If $f(x) = |\cos x - \sin x|$, then $f'\!\left(\dfrac{\pi}{6}\right)$ equals (give answer as integer after multiplying by $-1$ if negative):

If $y = \sqrt{\dfrac{1-x}{1+x}}$, then $\dfrac{dy}{dx}$ equals:

$f(x)=\begin{cases}\frac{\ln\cos x}{ax} & x>0\\ 0 & x=0\\ \frac{e^{x^2}-1}{bx} & x. If \(f'(0)=\frac{1}{4}$, then:

Which function is continuous everywhere in its domain but has at least one point where it is not differentiable?

Given function $f(x) = x^3 + x^2 f'(1) + xf''(2) + f'''(3)$. Then $f(2)$ equals:

Let $f: R \to R$ be differentiable at $c \in R$ and $f(c) = 0$. If $g(x) = |f(x)|$, then at $x = c$, $g$ is

\lim_{x \to \infty} \frac{\left(\sqrt{3x+1}+\sqrt{3x-1}\right)^6 + \left(\sqrt{3x+1}-\sqrt{3x-1}\right)^6}{\left(x+\sqrt{x^2-1}\right)^6 + \left(x-\sqrt{x^2-1}\right)^6} \cdot x^3

Given equation is $\ln(x+y) = 2xy$. Find $\frac{dy}{dx}$ at point $(0,1)$.

Given, $y = z^x$, then $y = z^y$. Find $\frac{dy}{dx}$.

\[\lim_{x \to 0} \frac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}\] is equal to: (where $[\,]$ denotes greatest integer function)

$\lim_{x \to 0} \frac{f(x) \cdot g(x)}{x(1-g(x))}$ will be

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)}, \quad 0

If $x + |y| = 2y$ then $y$ as a function of $x$ is

197. Let $g(x) = \dfrac{1}{f^{-1}(x)}$. Given the following data:$x$01234$f(x)$$-2$$-1$246$f'(x)$1/22/314/35/3The value of $g'(4)$ is:

If $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x\end{vmatrix}$, find $f'(1)$.

Let $f: R \to R$ be a function defined as \[f(x) = \begin{cases} 5, & \text{if } x \leq 1 \\ a + bx, & \text{if } 1

If the function \[f(x) = \begin{cases} x^3, & -2 \leq x where $f(x) = \sin(x-2) + a\cos(x-2)$is continuous and differentiable in $(4, 6)$, then find the range of $a$.

Let f(x) = P10 r=1 r sin rx x . The value of limx→0 f(x) is:

Let $f(x) = \dfrac{\tan x}{x}$, then the value of $\lim_{x \to 0}\left([f(x)] + x^2\right)^{\frac{1}{\{f(x)\}}}$ is equal to(where [.] and {.} denote greatest integer function and fractional part functions respectively)

Given limit = $\displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{f(\sec^2 x) \cdot 2\sec^2 x \tan x}{2x} = \dfrac{8}{\pi} f(2)$. If $k^a = 8^2$, find the value of $k^a$.

$f(x)=4|2x+3|+9[x+\frac{1}{2}]-12[x+20]$. Non-diff in (-20,20):

Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\dfrac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}$, $x\neq0$. If $L$ and $R$ respectively denote the left hand limit and the right hand limit of $f(x)$ at $x=0$, then $\dfrac{32}{\pi^2}(L^2+R^2)$ is equal to

The value of limn→∞(4n + 5n)1/n is:

55. If \(f(x) = \begin{cases} -e^{-x} + k, & x \leq 0 \\ e^x + 1, & 0

lim n→∞tan n X r=1 tan−1 1 1 + r + r2 ! is equal to:

Let $f(x) = x^3 - 9x^2 + 24x - 4$. Let $g(x)$ be defined as follows: \[g(x) = \begin{cases} f(x+2); & x If $g(x)$ is continuous for all $x$, find the value of $a$.

Let $u(x)$ and $v(x)$ be differentiable functions such that $\dfrac{u(x)}{v(x)} = 7$. If $\dfrac{u'(x)}{v'(x)} = p$ and $\left(\dfrac{u(x)}{v(x)}\right)' = q$, then $\dfrac{p+q}{p-q}$ has the value equal to:

30. Let $y$ be an implicit function of $x$ defined by $x^{2x} - 2x^x \cot y - 1 = 0$. The value of $y'(1)$, where $y'$ denotes the first derivative of $y$, is:

If the function $f(x)=\begin{cases}(1+|\cos x|)^{\frac{\lambda}{|\cos x|}}, & 0<x<\dfrac{\pi}{2}\\\mu, & x=\dfrac{\pi}{2}\\\dfrac{\cot 6x}{e^{\cot 4x}}, & \dfrac{\pi}{2}<x<\pi\end{cases}$ is continuous at $x=\dfrac{\pi}{2}$, then $9\lambda+6\log_e\mu+\mu^6-e^{6\lambda}$ is equal to:

Let $f(x) = \begin{cases} \left[1 + \ln(c^2 + c + 1)\tan^2(x-1)\right]^{\frac{1}{(\ln x)^2}}, & x \neq 1 \\ 3c, & x = 1 \end{cases}$, where $c \in R$.If $\lim_{x \to 1} f(x)$ exists but $f(x)$ is discontinuous at $x = 1$, then $c$ can take the value:

The function $f(x) = |x-3|, x \geq 1$\(\frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, x

199. Let $f(x)$ be a function defined by $f(x) = (k - x^{10})^{1/10}$ where $k = 1025$ and $f'(2) = \dfrac{1}{f'(a)}$ where $a \in N$, then $a$ equals:

The value of $\lim_{x \to 0} \left[\frac{3}{2} + \frac{x^2}{\sin x \tan x}\right]$ (where $[\cdot]$ denotes the greatest integer function) is

If $x = 3\tan t$ and $y = 3\sec t$, then the value of $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{\pi}{4}$ is:

Continuity Questions (1086)