Differentiability Questions - JEE

The value of $\lim_{x \to a} [2 - x + \sqrt{1 + x}]$, where $a \in [0, 1)$ and $[\cdot]$ denotes the greatest integer function is

The value of $\lim_{x \to 0} \frac{x \sin(\sin x) - \sin 2x}{x^6}$ equals

The value of $\lim_{n \to \infty} \frac{a^n + b^n}{a^n - b^n}$, where $a > b > 1$, is

Let f(x) = e^{x^3 + x^2 + x} for any real number x and let g be the inverse function for f. The value of g'(e^3) is

If f(x) = \begin{cases} \frac{\log(1+2ax)-\log(1+bx)}{x} & , x \neq 0 \\ k & , x = 0 \end{cases} is continuous at x=0, then k is equal to

Ex. 12 If $f(x) = [2 + 5|n|\sin x]$, where $n \in \mathbb{I}$ has exactly 9 points of non-derivability in $(0, \pi)$, then possible values of $n$ are (where $[x]$ denotes greatest integer function)

Let f : ℝ → ℝ is a function satisfying f(10 + x) = f(x) and f(2 + x) = f(2 − x), ∀x ∈ ℝ. If f(0) = 101. Then, the minimum possible number of values of x satisfying f(x) = 101, x ∈ [0, 25] is ……….

Let \[f(x) = \frac{e^{\tan x} - e^x}{\tan x - x} + \frac{\log(\sec x + \tan x) - x}{\tan x - x}\] be a continuous function at $x = 0$. The value of $f(0)$ equals

If $ g(x) = \begin{cases} \frac{a^x \log a + a^x}{x \log 2 + x \log a + 1}, & x \neq 0 \\ 2a + x \log 2 + x \log a + 1, & x = 0 \end{cases} $ where $a > 0$, then the value of $a$ for which $g(x)$ is continuous is:

Let $f : \mathbb{R} \to \mathbb{R}$ be such that $f(1) = 3$ and $f'(1) = 6$. Then, $\lim_{x \to 0} \frac{7f(1 + x)}{f(1)}$ equals [2002 AIEEE]

If f(x) = min {1, x², x³}, then

Let h(x) = min{x, x2} for every real number x. Then which of the following is true?

Let $m$ be a positive integer. If $\displaystyle\lim_{x \to 0} |\cos x + \sin 2x + \sin 3x|^{\cot x} = e^m$, then the value of $m$ is:

Let $\lim_{n \to \infty} \frac{1}{2} - \frac{1}{2} \left(1 + \frac{1}{n}\right)^{-1} \times \left(1^1 \times 2^2 \times 3^3 \times \ldots \times n^n\right)^{1/n^2} = e$, where $p$ and $q$ are relatively prime positive integers. Find the value of $|p + q|$.

The value of \[\lim_{x \to \infty} \frac{\int_0^x (\tan^{-1} x)^2 dx}{x^2 + 1}\]

The value of $\lim_{x \to -6} \frac{f(x) - x^2 - 1}{3(x + 6)}$ equals to:

Suppose that $f(0) = 0$ and $f'(0) = 2$. Let $g(x) = f(-f(-f(x)))$. The value of $g'(0)$ is:

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\begin{cases}x^2+3x+a & ,\; x\le 1\\bx+2 & ,\; x>1\end{cases}$ be differentiable on $\mathbb{R}$. Then the value of $\displaystyle\int_{-2}^{2}f(x)\,dx$ equals

Let $g(x)$ be a linear function and $f(x)=\begin{cases}g(x) & ,\; x\le 0\\\left(\dfrac{1+x}{2+x}\right)^{1/x} & ,\; x>0\end{cases}$ is continuous at $x=0$. If $f'(1)=f(-1)$, then the value of $g(3)$ is

If $y^2 = 3\cos^2 x + 2\sin^2 x$, then the value of $y^4 + y\frac{d^2y}{dx^2}$ is

If $f(x) = \begin{cases} \frac{(e^{x(x+3)\ln 27})^{\frac{x}{27}} - 9}{3^x - 27} & ; x 3 \end{cases}$and $\lim_{x \to 3} f(x)$ exists, then $l = $

The value of $\lim_{x \to -\infty} g(x)$ is:

If $f(x) = (x-1)^4(x-2)^3(x-3)^2$, then the value of $f'''(1) + f''(2) + f'(3)$ is:

Let $\lim_{n \to \infty} n \sin\left(\frac{2\pi e}{n}\right) = k\pi$, where $n \in \mathbb{N}$. Find k:

The value of $\displaystyle\lim_{x \to \frac{\pi}{2}} \dfrac{4(x-\pi)\cos^2 x}{\pi(\pi - 2x)\tan\!\left(x - \dfrac{\pi}{2}\right)}$ is equal to:

If $a=\displaystyle\lim_{x\to0}\dfrac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $b=\displaystyle\lim_{x\to0}\dfrac{\sin^2x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $ab^3$ is:

If $\lim_{x \to \infty} \frac{px + q}{qx + p} = 1$ and $\lim_{x \to \infty} \frac{px + q}{qx + p} = m$, where $p, q \neq 0$, then $\lim_{x \to 0}$ is

If $e^y + xy = e$, the ordered pair $\!\left(\dfrac{dy}{dx},\,\dfrac{d^2y}{dx^2}\right)\!$ at $x=0$ is:

If $G(x) = -\sqrt{25 - x^2}$ then $\lim_{x \to 1} \frac{G(x) - G(1)}{x - 1} =$ ______

If $2x = y^{1/5} + y^{-1/5}$ and $(x^2 - 1)\dfrac{d^2y}{dx^2} + \lambda x\dfrac{dy}{dx} + ky = 0$, then $\lambda + k$ is equal to

We have $\lim_{x \to 1} \dfrac{x^4 - 1}{x - 1} = \lim_{x \to k} \dfrac{x^3 - k^3}{x^2 - k^2}$. Find the value of $k$.

$\lim_{t \to 0} \lim_{x \to \infty} \cot\left(\frac{\pi(1 - t^2f(x) \cdot g(x))}{4}\right)$ is equal to

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

lim x→0+ cos−1(x −[x]2) · sin−1(x −[x]2) x −x3 , where [x] denotes the greatest integer less than or equal to x, is:

If $\lim_{x \to 0}\left(1 + \int_0^{\sqrt{a^x-1}} (\sin(2\,\text{arc}\tan t))(1+t^2)^{\ln a}\,dt\right)^{\!\frac{1}{x}} = 5$, then find the value of $a$, where $a \in N$ and $a > 1,\; x > 0$.

If $\lim_{n \to \infty} \frac{\sqrt[n]{(2n)^n}}{n}$ is equals to $(a-1)e^b$, then

Find: $\lim_{b \to 0} \frac{r_{max} - r}{\sin b}$

Let $f(x) = \displaystyle\lim_{n \to \infty} \dfrac{x^{2n-1} + ax^3 + bx^2}{x^{2n} + 1}$ is continuous for all $x \in \mathbb{R}$. If points $A(-a, 3)$ and $B((b+1), -1)$ are points of relative maximum and minimum of a cubic polynomial $y = g(x)$, then the value of $g(2)$ is:

Let f(x) = 1−x(1+|1−x|) |1−x| cos 1 1−x for x ̸= 1. Find LHL and RHL at x = 1.

Consider a parabola $y = \frac{x^2}{4}$ and the point $F(0,1)$. Let $A_1(x_1, y_1), A_2(x_2, y_2), A_3(x_3, y_3), \ldots, A_N(x_n, y_n)$ are 'n' points on the parabola such that $x_k > 0$ and $\angle OFA_k = \frac{k\pi}{2n}$ $(k = 1,2,\ldots,n)$. If the value of $\lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} FA_k = \frac{m}{\pi}$, then $m$ is ______.

Suppose $x_1 = \tan^{-1}2 > x_2 > x_3 > \ldots$ are the real numbers satisfying $\sin(x_{n+1} - x_n) + 2^{-(n+1)} \sin x_n \sin x_{n+1} = 0$ for all $n > 1$ and the sequence is convergent and $l = \lim_{n \to \infty} x_n$, the value of $4l$ is ______.

Let $H_n$ denotes the harmonic mean of $n$ positive integers $n+1, n+2, n+3, \ldots, n+n$. If $\lim_{n \to \infty} \left( \frac{H_n}{n} \right) = \frac{1}{k}$ then the value of $e^k$ is ______.

Let $f:\mathbb{R}\to(0,\infty)$ be a twice differentiable function such that $f(3)=18$, $f'(3)=0$ and $f''(3)=4$. Then $\displaystyle\lim_{x\to1}\left(\log_e\left(\dfrac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^2}}\right)$ is equal to:

The value of $\displaystyle\lim_{x\to0}\frac{\log_e(\sec(ex)\cdot\sec(e^2x)\cdots\sec(e^{10}x))}{e^2-e^{2\cos x}}$ is equal to

If $f(x) = x^4 \tan x^3 - x\ln(1 + x^2)$, then the value of $\dfrac{d^4 f(x)}{dx^4}$ at $x = 0$ is:

Let $ f(x) $ be defined as $ f(x) = kx + 2 $ for $ x \leq -1 $ and $ f(x) = 2x + 3 $ for $ x > -1 $. If $ f $ is continuous at $ x = -1 $, then $ k $ equals

Let $f(x) = \begin{cases}(x+1)(x+2), & x > 0 \\ a\sin x + b\cos x, & x \leq 0\end{cases}$. If f is differentiable at x = 0, find the value of a − b.

If $f(x)$ be such that $f(x) = \max(|3-x|, 3-x^3)$ then:

If $\lim_{x \to 0} \frac{x(1 + a\cos x) + b\sin x}{x^3} = 1$ then

If $e^y + xy = e$, the ordered pair $\left(\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}\right)$ at $x = 0$ is equal to:

Differentiability Questions (1063)