Continuity Questions (1086)

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:

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 $x=\tfrac{1}{2}$: