Differentiability Questions (1063)

155. The value of \(\displaystyle\lim_{x \to 0} \dfrac{\dfrac{x^2}{2} + 1 - \sqrt{1 + x^2}}{\left(\cos x - e^{x^2}\right)\sin(x^2)}\) is equal to:

27. Consider a function \(f: R \to R\) such that \(f(x) = \begin{cases} \sin(\pi x), & \text{if } x \in \mathbb{Q} \\ \tan(\pi\sqrt{|x|}), & \text{if } x \notin \mathbb{Q} \end{cases}\). If \(\displaystyle\lim_{x \to N} f(x)\) exists, then the sum of all positive integers \(N < 100\), is equal to:

Let $f:\mathbb{R}-\{0\}\to\mathbb{R}$ be a function satisfying $f\left(\dfrac{x}{y}\right)=\dfrac{f(x)}{f(y)}$ for all $x,y$, $f(y)\ne 0$. If $f'(1)=2024$, then

Let $f(x)=\sqrt{\displaystyle\lim_{r\to x}\left\{\frac{2r^2[(f(r))^2-f(x)f(r)]}{r^2-x^2}-r^3 e^{f(r)/r}\right\}}$ be differentiable in $(-\infty,0)\cup(0,\infty)$ and $f(1)=1$. Then the value of $ea$, such that $f(a)=0$, is equal to ______.

Let $f:[-1,2]\to\mathbb{R}$ be given by $f(x)=2x^2+x+[x^2]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is:

If $\alpha=\displaystyle\lim_{x\to0^+}\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}$ and $\beta=\displaystyle\lim_{x\to0}(1+\sin x)^{\frac{1}{2}\cot x}$ are the roots of the quadratic equation $ax^2+bx-\sqrt{e}=0$, then $12\log_e(a+b)$ is equal to

If $y = \tan^{-1}\!\left(\dfrac{6x-4-4x^2}{1+6x^2+8x^3}\right)$ and $\dfrac{dy}{dx} = \dfrac{A}{1+4x^2}+\dfrac{B}{1+x^2}$... find $24(A+B)$. [Integer type]

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:

The value of $\displaystyle\lim_{x\to0}2\left(\frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}\cdots\sqrt[10]{\cos 10x}}{x^2}\right)$ is

Let $f:\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\to\mathbb{R}$ be a differentiable function such that $f(0)=\dfrac{1}{2}$. If $\displaystyle\lim_{x\to0}\dfrac{x\int_0^x f(t)\,dt}{e^{x^2}-1}=\alpha$, then $8\alpha^2$ is equal to:

The value of \(\lim_{x \to \frac{\pi}{4}} \frac{\int_2^{\csc^2 x} g(t)dt}{x^2 - \frac{\pi^2}{16}}\) is:

If $\displaystyle\lim_{x\to1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\frac{m\sqrt{5}}{n(2n)^{2/3}}$, where $\gcd(m,n)=1$, then $8m+12n$ is equal to

\[\lim_{n \to \infty} \frac{e^n}{\left(1 + \dfrac{1}{n}\right)^{n^2}}\] equals

If \(f(x)\) is a polynomial of least degree such that \(\lim_{x \to 0}\left(1 + \dfrac{f(x) + x^2}{x^2}\right)^{1/x} = e^2\), then \(f(2)\) is

Let \(f_n(x) + f_n(y) = \frac{x^n + y^n}{x^n y^n}\) for all \(x, y \in \mathbb{R} - \{0\}\) where \(n \in \mathbb{N}\).Let \(g(x) = \max\left\{f_2(x), f_3(x)\right\}\) for all \(x \in \mathbb{R} - \{0\}\).The number of values of \(x\) for which \(g(x)\) is non-differentiable (\(x \in \mathbb{R} - \{0\}\)):

The value of limx→∞  x2−2x+1 x2−4x+2 x is:

The value of \(\displaystyle\lim_{x \to 0} \dfrac{\dfrac{x^2}{2} + 1 - \sqrt{1+x^2}}{\left(\cos x - e^{x^2}\right)\sin(x^2)}\) is equal to:

If composite function \(f_1(f_2(f_3(\ldots(f_n(x))\ldots)))\) (n times) is an increasing function and if r of \(f_i\)'s are decreasing functions while rest are increasing, then maximum value of function is

If the function $f(x) = \begin{cases} \frac{2}{x}\{\sin(k_1+1)x + \sin(k_2-1)x\}, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x}\log_e\!\left(\frac{2+k_1x}{2+k_2x}\right), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to:

Let the function f (x) = (x + 1) ∣∣x - ax + 2∣∣ + cos |x| be not differentiable at the two points x = \alpha = 2 and 2 2 x = \beta . Then the distance of the point (\alpha, \beta) from the line 12x + 5y + 10 = 0 is equal to :

Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f (x) = [x] + |x - 2|, -2 < x < 3, is not continuous and not differentiable. Then m + n is equal to :

If \(x = 4t^3\) and \(y = 3t^4\) (parametric form), and \(\dfrac{d^2x/dy^2}{(dx/dy)^n}\) = constant, find n. Then compute the sum \(\dfrac{4/5}{1 - 1/n}\).

Let $f(x)=\displaystyle\int_0^x(t+\sin(1-e^t))\,dt$, $x\in\mathbb{R}$. Then $\displaystyle\lim_{x\to0}\frac{f(x)}{x^3}$ is equal to

Let the function $f(x) = (x^2+1)|x^2 - ax + 2| + \cos|x|$ be not differentiable at the two points $x = \alpha = 2$ and $x = \beta$. Then the distance of the point $(\alpha, \beta)$ from the line $12x + 5y + 10 = 0$ is equal to:

If $\lim_{x \to \infty}\!\left(\!\left(\frac{e}{1-e}\right)\!\left(\frac{1}{e} - \frac{x}{1+x}\right)\!\right)^x = \alpha$, then the value of $\dfrac{\log_e \alpha}{1 + \log_e \alpha}$ equals:

Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbb{N}$ for which $\lim_{x \to 0^+}\!\left(x\!\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \cdots + \left[\frac{p}{x}\right]\right) - x^2\!\left(\left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \cdots + \left[\frac{9^2}{x^2}\right]\right)\right) \geq 1$ is equal to ___

Suppose the function $f(x) - f(2x)$ has the derivative 5 at $x = 1$ and derivative 7 at $x = 2$. The derivative of the function $f(x) - f(4x)$ at $x = 1$ has the value equal to:

If $y = 2 + \sqrt{\sin x + 2 + \sqrt{\sin x + 2 + \sqrt{\sin x + ...\infty}}}$, then the value of $\frac{dy}{dx}$ at $x = 0$ is

Let \(x_n\) be a positive root of the equation \(x^n = x^2 + x + 1\). Then the value of \(e^{\left(\lim_{n\to\infty} n(x_n - 1)\right)}\) is:

If $x = 2\cos t - \cos 2t$ and $y = 2\sin t - \sin 2t$, then $\dfrac{dy}{dx}$ at $t = \dfrac{\pi}{2}$ equals:

Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f:[0,\infty)\to\mathbb{R}$ be a function defined by $f(x)=\left[\dfrac{x}{2}+3\right]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which $f$ is not continuous. Then $\sum_{a\in S}a$ is equal to ________.

Ex. 38: Statement I: If $\lim_{x \to 0} \frac{f(x)}{\sin x}$ does not exist, then $\lim_{x \to 0} f(x)$ does not exist.Statement II: $\lim_{x \to 0} \frac{e^{1/x} - 1}{e^{1/x} + 1}$ does not exist.

For \( x \in R \), \( f(x) = |\log 2 - \sin x| \) and \( g(x) = f(f(x)) \), then:

Given \(f(x) = \tan^{-1}\left(\dfrac{\sin x - \cos x}{\sin x + \cos x}\right)\), then \(\dfrac{df(x)}{dx}\) equals:

If \(\lim_{x \to s} f(x)\) and \(\lim_{x \to s} g(x)\) exist then \(\lim_{x \to s} \frac{f(x)}{g(x)}\) exists.

If \(a^2 + b^2 + c^2 + ab + bc + ca \leq 0\), where \(a, b, c \in R\) and \(f(x) = a[x] + b|x| + c\,\text{sgn}(x)\), then in \((-2, 2)\), which of the following is not true?[Note: \([y]\) denotes greatest integer function less than or equal to \(y\).]

The quadratic equation whose roots are the minimum value of $$\sin^{-1}\frac{2}{\sqrt{5}} - \sin^{-1}\frac{1}{\sqrt{5}}$$ and $$\lim_{x \to \infty} \frac{(x+1)(x+2) - x}{2}$$ is

Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=x^3+x-5$. If $g(x)$ is a function such that $f(g(x))=x$ for all $x\in\mathbb{R}$, then $g'(105)$ is equal to:

The value of \(\lim_{x \to 0} \frac{e^{(1+x)^{1/x}} - e}{\tan x}\) is

Let \[ f(x) = \begin{cases} xe^{-\left(\frac{1}{|x|}+\frac{1}{x}\right)}, & x \neq 0 \\ 0, & x = 0 \end{cases} \] Then \(f(x)\) is

If A = limx→0 sin−1(sin x) cos−1(cos x) and B = limx→0 [|x|] x , then:

\(f(x)=\begin{cases}\max\{|x|,x^2\} & |x|\le 2\\ 8-2|x| & 2. S = non-diff in (-4,4).

Evaluate \(\lim_{x \to 0} \dfrac{64^x - 32^x - 16^x + 4^x + 2^x - 1}{\left[\sqrt{(15 + \cos x)} - 4\right]\sin x}\)

\(f(x+y)=f(x)+f(y)+xy^2+x^2y\), \(\lim_{x\to 0}f(x)/x=1\). Find \(f'(3)\).

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

If \( f'(x) + (f'(x))^2 + (f'(x))^3 + (f'(x))^4 + \cdots \infty = e^x \), where \( f'(x) \in (-1, 1) \) and \( f(0) = 0 \), then the value of \( \lim_{x \to 0} (1 + f(x))^{\frac{1}{x}} \) is equal to:

⎧ ⎪ 3x, x < 0 Let f (x) = ⎨ min{1 + x + [x], x + 2[x]}, 0 \le x \le 2 ⎩ ⎪ 5, x > 2, where [.] denotes greatest integer function. If \alpha and \beta are the number of points, where f is not continuous and is not differentiable, respectively, then \alpha + \beta equals __________

Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x+y) = f(x)f'(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\displaystyle\sum_{n=1}^{100} \log_e f(n)$ is equal to:

Let $a$ be the sum of all coefficients in the expansion of $(1-2x+2x^2)^{2023}(3-4x^2+2x^3)^{2024}$ and $b=\displaystyle\lim_{x\to0}\left(\dfrac{\int_0^x\dfrac{\log(1+t)}{t^{2024}+1}\,dt}{x^2}\right)$. If the equations $cx^2+dx+e=0$ and $2bx^2+ax+4=0$ have a common root, where $c,d,e\in\mathbb{R}$, then $d:c:e$ equals

Let $f:\mathbb{R}\to\mathbb{R}$ be a function given by $f(x)=\begin{cases}\dfrac{1-\cos2x}{x^2}, & x<0\\ \alpha, & x=0\\ \dfrac{\beta\sqrt{1-\cos x}}{x}, & x>0\end{cases}$ where $\alpha,\beta\in\mathbb{R}$. If $f$ is continuous at $x=0$, then $\alpha^2+\beta^2$ is equal to: