Differentiability Questions (1063)

If \(\lim_{x \to 1} \dfrac{x^4 - 1}{x - 1} = \lim_{x \to k} \dfrac{x^3 - k^3}{x^2 - k^2}\), then \(k\) is __________ (up to two decimal places).

\(\lim_{x \to \infty} \left(\dfrac{x^2 + 5x + 3}{x^2 + x + 2}\right)^x\) is equal to

If $f(x) = \sqrt{x - 4\sqrt{x - 4}} + \tan^{-1}\left(\frac{1+x}{1-x}\right), 4 < x < 8$, then the value of $f'(5)$ is equal to

\(\lim_{x \to 0} \dfrac{(1 - \cos 2x)^2}{2x \tan x - x \tan 2x}\) is

If $f(x) = |x|^{\{x\}} + (\{x\})^2 + \sin(\pi x)$, where $[\cdot]$ and $\{\cdot\}$ represent the greatest integer function and the fractional part function respectively, then $f'\left(\frac{3}{2}\right)$ is

Let $f:(-2,2)\to\mathbb{R}$ be defined by $f(x)=\begin{cases}x[x], & -2<x<0\\ (x-1)[x], & 0\leq x<2\end{cases}$ where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2,2)$ at which $y=|f(x)|$ is not continuous and not differentiable, then $m+n$ is equal to ________.

Let $f(x)$ be a differentiable function on $x \in R$, such that $f(x + y) = f(x)f(y)$ for all $x, y \in R$ where $f(0) \neq 0$. If $f'(5) = 10$, $f'(0) = 0$, then the value of $f'(5)$ is equal to

Given that \( f(x) = \dfrac{\sqrt{2}\cos x - 1}{\cot x - 1} \) is continuous at \( x = \dfrac{\pi}{4} \). Find the value of \( f\!\left(\dfrac{\pi}{4}\right) = k \).

If \(x^{2x} - 2x^x \cot y - 1 = 0\), find \(y'(1)\).

In $(0, 2\pi)$, the total number of points where $f(x) = \max(\sin x, \cos x, 1 - \cos x)$ is not differentiable, are equal to

The number of points at which the function $f(x) = |x - 0.5| + |x - 1| + \tan x$ is not differentiable in the interval $(0, 2)$ is/are

For $a,b>0$, let $f(x)=\begin{cases}\dfrac{\tan((a+1)x)+b\tan x}{x}, & x<0\\ 3, & x=0\\ \dfrac{\sqrt{ax+b^2x^2}-\sqrt{ax}}{b\sqrt{ax}\sqrt{x}}, & x>0\end{cases}$ be a continuous function at $x=0$. Then $\dfrac{b}{a}$ is equal to:

\(\lim_{x \to 0} \dfrac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}\) equals __________ (up to four decimal places).

Consider the function $f(x) = \max\{\sin x, |\cos x|\}, \forall x \in [0, 2\pi]$. If $n$ is the number of points at which $f(x)$ is non-differentiable, then the value of $n$ is

If $f(x) = \begin{cases} \frac{x \ln(e + \cos x)}{x^2} & x > 0 \\ q & x \leq 0 \end{cases}$ is continuous at $x = 0$, then the value of $\frac{pq+1}{q}$ is

If $y = \tan^{-1}\left(\frac{1}{x^2+1}\right) + \tan^{-1}\left(\frac{1}{x^2+3}\right)$, find $\frac{dy}{dx}$

If \(f(n+1) = \dfrac{1}{2}\left\{f(n) + \dfrac{9}{f(n)}\right\}\) where \(n \in N\) and \(f(n) > 0\ \forall n \in N\) and \(\underset{n\to\infty}{\text{Lt}}\ f(n)\) exist then the value of \(\underset{n\to\infty}{\text{Lt}}\ f(n) =\)

If \(f(x)\) is a real valued bijective function satisfying \(f'(x) = \sin^2(\sin(x+1))\) and \(f(0) = 3\), then the value of \((f^{-1})''(3)\) is equal to:

Let f be a real valued derivable function such that f(x)f(y) = f(x)y + xf(y), ∀x, y ∈ ℝ. If f′(0) = 2, then find \(\lim_{x \to 0} \left[\dfrac{f(x)}{\sin x}\right]\). [Note: [ ] represents greatest integer function.]

Let $f$ and $g$ be two functions defined by $f(x)=\begin{cases}x+1, & x<0\\|x-1|, & x\geq0\end{cases}$ and $g(x)=\begin{cases}x+1, & x<0\\1, & x\geq0\end{cases}$. Then $(g\circ f)(x)$ is

Let $f(x)=[x^2-x]+|-x+[x]|$, where $x\in\mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is

Let \( f(x) = \dfrac{1 - \tan x}{4x - \pi} \), \( x \neq \dfrac{\pi}{4} \), \( x \in \left[0, \dfrac{\pi}{2}\right] \). If \( f(x) \) is continuous in \( \left[0, \dfrac{\pi}{2}\right] \), then \( f\!\left(\dfrac{\pi}{4}\right) \) is

If lim x\to\infty (( e )( 1 - x )) = \alpha , then the value of log e \alpha equals : 1-e e 1+x 1+log \alpha e

If the function 2 ⎧ {sin(k1 + 1)x + sin(k2 - 1)x} , x < 0 ⎪ ⎪ x ⎪ 4, x = 0 f (x) = ⎨ ⎪ ⎪ 2 2+k1 x ⎩ ⎪ log ( ), x > 0 x e 2+k2 x is continuous at x = 0, then k + k is equal to 2 1 2 2

$\displaystyle\lim_{n\to\infty}\left\{\left(2^{1/2}-2^{1/3}\right)\left(2^{1/2}-2^{1/5}\right)\cdots\left(2^{1/2}-2^{1/(2n+1)}\right)\right\}$ is equal to

Evaluate $\lim_{x \to \frac{\pi}{2}} \frac{\tan\left(\frac{x}{2}\right)(1-\sin x)}{(x-2y)^3}$

We have \[\lim_{x \to 1} \left(1 + \frac{a}{x} - \frac{4}{x^2}\right)^{2x} = e^3\] Find the value of \(a\).

If \(y = e^{ax^{-4x}}\) and \(z = e^{-\ cos^{-1}x}\) then \(\dfrac{d^2y}{dz^2} = 0\).State whether the statement is true or false.

Find: \(\lim_{n \to \infty} \left(\frac{1^{1/x} + 2^{1/x} + 3^{1/x} + \cdots + n^{1/x}}{n}\right)^{nx}\)

If \(f(1) = g(1) = 2\) and \(f'(1)\), \(g'(1)\) exist, then evaluate \[\lim_{x \to 1} \frac{f(1)g(x) - f(x)g(1)}{g(x) - f(x)}\]

$\lim_{x \to 0} \frac{2e^{\sin x} - e^{-\sin x} - 1}{x^2 + 2x}$ equals:

If \(g(x) = (x^2 + 2x + 3)f(x)\), \(f(0) = 5\) and \(\displaystyle\lim_{x \to 0}\left(\frac{f(x) - 5}{x}\right) = 4\), then \(g'(0)\) is equal to:

The value of $\lim_{x \to 0} \frac{\sin x + \cos(e^x)}{1 + \sin(3x)}$ is equal to

Let $f(x) = 10 - |x - 5|, x \in \mathbb{R}$, then the set of all values of $x$ at which $f(f(x))$ is not differentiable is

Let $f(x) = \sin(\cos^{-1}(\sin x))$ and $g(x) = \cos(\sin^{-1}(\cos x))$. If $S$ is the range of $\dfrac{f'(x)}{g'(x)}$, then $S$ contains:

If $\displaystyle\lim_{x\to0}\dfrac{e^{ax}-\cos(bx)-cxe^{-cx}}{1-\cos(2x)}=17$, then $5a^2+b^2$ is equal to

$\displaystyle\lim_{x\to0}\frac{e-(1+2x)^{\frac{1}{2x}}}{x}$ is equal to

Let $a\in\mathbb{Z}$ and $[t]$ be the greatest integer $\leq t$, then the number of points, where the function $f(x)=[a+13\sin x]$, $x\in(0,\pi)$ is not differentiable, is ____________.

If \(\lim_{x \to 1} \dfrac{x^2 - ax + b}{x - 1} = 5\), then \(a + b\) is equal to __________.

For each t ∈ R, let [t] be the greatest integer less than or equal to t. Then limx→0+ x  1 x  +  2 x  + · · · + 15 x 
: (1) is equal to 15. (2) is equal to 120. (3) does not exist. (4) is equal to 0.

If \(f(x) = \{x + \sin x\} + [x - \sin x] + [x]\) where \([y]\) and \(\{y\}\) denote greatest integer function and fractional part function of \(y\) respectively, then find the number of points of discontinuity in \([0, \pi]\).

If $f(x) = \begin{cases} \sqrt{t+x^2} & x > 0 \\ a & x = 0 \text{ is continuous at } x = 0 \text{ for some constants } a, b \text{ and } c, \text{ then the value of } \frac{abc}{b^2} \text{ is equal to} \\ \frac{c^2+x}{b^2+x^2} & x < 0 \end{cases}$

\(\lim_{x \to 0} \dfrac{(1 - \cos 2x)(3 + \cos x)}{x \tan 4x}\) is equal to

Let f : R \rightarrow R be a positive increasing function with \( \lim_{x \to 0} \frac{f(3x)}{f(x)} = 1 \). Then \( \lim_{x \to 0} \frac{f(2x)}{f(x)} = \)

Let \(f(1^+) = f(1) = f(1^-)\) and \(f'(1^-) = f'(1^+)\). Given \(f(x) = a + \cos^{-1}(x+b)\) (for \(x \geq 1\)) and \(f(x) = -\dfrac{1}{\sqrt{1-(1+b)^2}}\) type condition at \(x=1\), find \(\dfrac{a}{b}\).

Set of a for which limx→a([x −5] −[2x + 2]) = 0:

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

\(f=\min\{1,1+x\sin x\}\) on \([0,2\pi]\). \((m,n)=\)

If \( f : R \to R \) is a function defined by \[ f(x) = [x]\cos\left(\frac{2x-1}{2}\right)\pi, \] where \( [x] \) denotes the greatest integer function, then \( f \) is

If \( \lim_{x \to 0} \dfrac{10 - \displaystyle\sum_{k=1}^{10}(\cos kx)}{x^2} = \dfrac{a}{b} \) where a and b are co-prime, then the value of \( (a + b) \) is equal to: