Continuity Questions (1086)

If $f(x) = \begin{cases} 2x^2 + 3 & x > 3 \\ ax^2 + bx + 1 & x \leq 3 \end{cases}$ is differentiable everywhere, then $a$ is equal to

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 \(k\).]

The function $f(x) = \max\{(1-x), (1+x), 2\}$ $\forall x \in \mathbb{R}$ is

If $y = \tan^{-1}\left(\frac{1}{1+x^2}\right) + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$ ($y > 0$), then $\frac{dy}{dx}$ is equal to

If $a = \sec f$ and $b = \csc f$, where $f$ is a parameter, then the value of $\frac{dy}{dx}$ when $x = -\frac{3}{4}$ is

If $y = (1+x^2)^n + \sin^{-1}(\sin^2 x)$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is

Let \(f(x) = x\left[\frac{x}{2}\right]\), for \(-10

Let h(x) = 2 − |x − 1| and g(x) = h(|x|) + |h(x)|. Find the number of points where g(x) is non-differentiable.

The value of $\lim_{x \to \infty} \frac{ax^2 + bx + c}{dx + e}$ ($a, b, c, d, e \in \mathbb{R} - \{0\}$) depends on the sign of:

If $f(x) = 2x^3 - 3x^2 + a$ is a one-to-one function, the value of $a$ is arbitrary. Given $f(g(x))=x$, then $g'\!\left(\dfrac{11}{4}\right)$ is:

\[ \lim_{x \to 0} \frac{\sin{(\pi \cos{x})}}{x^2} \]

Find $\frac{dy}{dx}$ at $x = 3$ where $y = \sqrt{x^2 + 16}$ and $s = \frac{x}{y}$

If $x = \frac{1-\tan^2\theta}{1+\tan^2\theta}$ and $y = \frac{2\tan\theta}{1+\tan^2\theta}$, find $\frac{dy}{dx}$

25. If \(y = (x + \sqrt{1 + x^2})^n\), then the value of \((1 + x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}\) is equal to:

Consider the function $f(x) = \tan^{-1}\left(\frac{x^2-1}{x^2+1}\right), \forall x \geq 0$. If $g(x)$ is the inverse function of $f(x)$, then the value of $g'\left(\frac{\pi}{4}\right)$ is equal to

The integer \(n\) for which \(\lim_{x \to 0} (\sin x)^{1/x}\) is a finite non-zero number, is [2002 AIEEE]

If $I_n = \frac{d^n}{dx^n}(x^n\ln x)$, then the value of $\frac{d}{dx}(I_7 - 7I_6)$ is equal to

198. If \(y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}\), then the value of \(\dfrac{dy}{dx}\) at \(x = 8\) is:

The total number of points of non-differentiability of \(f(x) = \max\left\{\sin^2 x, \cos^2 x, \frac{3}{4}\right\}\) in \([0, 10\pi]\), is

If $\lim_{x \to \frac{\pi}{2}} \frac{\sin x}{x}$ exists finitely, then the value of $a$ is

The value of $\lim_{x \to \pi} \left[1 - \cos\left(\frac{x}{2}\right) - \cos\left(\frac{x}{4}\right) + \cos\left(\frac{x}{2}\right) \cdot \cos\left(\frac{x}{4}\right)\right]$ is $\frac{\lambda}{4}$, then the value of $900\lambda$ is equal to there, $\lambda > 0$)

If $\lim_{x \to 0} \frac{\sin 3x}{\sin 4x} - 3$, then the value of $272 \cdot \frac{ab}{cd}$ is equal to

The value of $\lim_{x \to 0} \frac{\sin\left(\frac{x}{3}\right) \sin\left(\frac{x}{3}\right)}{\left(\frac{x}{3}\right)}$ is equal to

If $x = t^3 + t^5$ and $y = \sin t$, then $\dfrac{d^2y}{dx^2}$ equals:

Let $f(x) = x^2 - 4x - 3, x > 2$ and $g(x)$ is the inverse of $f(x)$. Then the value of $\frac{1}{g'(2)}$, where $f(z) = 2$ is (here, $g'$ represents the first derivative of $g$)

Find \(\displaystyle\lim_{x \to \infty} \frac{\displaystyle\int_0^x e^{t^2}\, dt}{\displaystyle\int_0^x e^{2t^2}\, dt}\).

198. If \(y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}\), then the value of \(\dfrac{dy}{dx}\) at \(x = 8\) is:

Find $\lim_{x \to 4} \frac{(1-\sin(\pi x))}{(x-4)^2}$

If \(y = 1 + \dfrac{c_1}{x - c_1} + \dfrac{c_2 x}{(x - c_1)(x - c_2)} + \dfrac{c_3 x^2}{(x - c_1)(x - c_2)(x - c_3)}\), then \(\dfrac{dy}{dx}\) is equal to:

Let \(f\) be continuous on \([a, b]\), differentiable on \((a, b)\), and \(f(a)/a = f(b)/b\). Then by Rolle's theorem applied to \(g(x) = f(x)/x\), there exists \(x_0 \in (a, b)\) such that:

196. Let f and g be defined such that \(f'(x) = f^2(x) + g^2(x)\) and \(g'(x) = 2f(x)g(x) + 1\). If \(f(0) = \dfrac{1}{5}\), \(g(0) = \dfrac{4}{5}\), then the value of \(f\!\left(\dfrac{\pi}{12}\right) + g\!\left(\dfrac{\pi}{12}\right)\) equals:

Consider the following statements about positive functions \(f(x)\) and \(g(x)\) whose limits to infinity exist.Statement-A: \(\lim_{x\to\infty} f(x) = \lim_{x\to\infty} g(x)\)Statement-B: \(\lim_{x\to\infty}(f(x)-g(x))=0\)Statement-C: \(\lim_{x\to\infty}\sqrt{f(x)} = \lim_{x\to\infty}\sqrt{g(x)}\)How many of the following six statements are true: \(A\Rightarrow B,\; B\Rightarrow C,\; C\Rightarrow A,\; A\Rightarrow C,\; B\Rightarrow A,\; C\Rightarrow B\)?

The value of $$\lim_{x \to 0^+} \frac{\int_1^{\cos x} (\cos^{-1} t) \, dt}{2x - \sin 2x}$$ is equal to:

If \(x = e^t \cos t\), \(y = e^t \sin t\) then \(\dfrac{d^2 y}{dx^2}\) = ______

Let p = \( \lim_{x \to 0^+} (1 + \tan^2{\sqrt{x}})^{\frac{1}{x}} \) then log p is equal to -

Let \(f(x) = x \cos x + 2\) and \(g(x)\) be the inverse function of \(f(x)\), then \(g'(3)\) equals to ……… .

If \(\displaystyle\lim_{\alpha\to 0} \frac{e^{\cos(\alpha^n)} - e}{\alpha^m} = \frac{-e}{2}\) where \(m\) and \(n\) are positive integers greater than 1, then the value of \(\dfrac{m}{n}\) is:

\(\lim_{x \to 0} \dfrac{x \cot(4x)}{\sin^2 x \cot^2(2x)}\) is equal to __________.

797. Let \(f(x) = x\), \(g(x) = |1 - f(x)|\), \(h(x) = 2 - g(x)\), \(L(x) = h(|x|) + |h(x)|\). Find the number of points where \(L(x)\) is non-differentiable.

The number of values of \(x\), \(x \in [-2, 3]\) where \(f(x) = [x^2]\sin(\pi x)\) is discontinuous is (where \([\cdot]\) denotes greatest integer function)

If f(x) = \(\begin{cases} \frac{-x}{2}, & x > 0 \\ -\cos x, & 0 \leq x \leq 1 \\ \ln x, & x > 1 \end{cases}\), then find the number of points where f(x) is not differentiable.

\(\lim_{x \to 0} \dfrac{x + 2\sin x}{\sqrt{x^2 + 2\sin x + 1} - \sqrt{\sin^2 x - x + 1}}\) is __________.

If \( f(x) = \begin{cases} ax + b, & x \le -1 \\ ax^3 + x + 2b, & x > -1 \end{cases} \) is differentiable for all \( x \in \mathbb{R} \), then find \( b - a = \) __________.

Let \(f : R \to R\) be a continuously differentiable function such that \(f(2) = 6\) and \(f'(2) = \dfrac{1}{48}\). If \(\displaystyle\int_0^{f(x)} 4t^3\, dt = (x - 2)\, g(x)\), then \(\lim_{x \to 2} g(x)\) is equal to __________.

Find $\lim_{x \to \infty} \left( \frac{p^{1/x} + q^{1/x} + r^{1/x}}{3} \right)^h$ [where $p, q, r, a > 0]$

The value of $f(0)$ so that the function $f(x) = \frac{1 - \cos(1 - \cos x)}{x^4}$ is continuous everywhere is $k$, then value of $10k$ is

\(\lim_{x \to \pi/4} \dfrac{\cot^3 x - \tan x}{\cos(x + \pi/4)}\) is __________.

200. If \(y = 2\tan^{-1}\!\left(\dfrac{\sqrt{1+x^2}-1}{x}\right)\), then the value of \(\dfrac{d^2y}{dx^2}\) at \(x = 2\) is:

$\displaystyle\lim_{x\to\frac{\pi}{2}}\frac{\displaystyle\int_{x^3}^{(\pi/2)^3}(\sin(2t^{1/3})+\cos(t^{1/3}))\,dt}{\left(x-\dfrac{\pi}{2}\right)^2}$ is equal to

Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=\displaystyle\lim_{p\to\infty}\frac{\sin\!\left((2x-1)\frac{\pi}{2}\right)\!x^{4p}e^{x^2-1}+x^{4p}}{1+x^{4p+2}-x^{4p}}$. $f(x)$ is continuous for all $x$ in