Limits, Continuity & Differentiability Questions - JEE

Let the function $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ a^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1$, $b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta\sqrt{3}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ___

Let $a>0$ be a root of the equation $2x^2+x-2=0$. If $\displaystyle\lim_{x\to\frac{1}{a}}\frac{16(1-\cos(2+x-2x^2))}{(1-ax)^2}=\alpha+\beta\sqrt{17}$, where $\alpha,\beta\in\mathbb{Z}$, then $\alpha+\beta$ is equal to

If the function $f(x)=\begin{cases}\dfrac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x\neq0\\ a\log_e2\log_e3, & x=0\end{cases}$ is continuous at $x=0$, then the value of $a^2$ is equal to:

Let $f:(0,\pi)\to\mathbb{R}$ be a function given by $f(x)=\begin{cases}\left(\dfrac{8}{7}\right)^{\frac{\tan8x}{\tan7x}}, & 0<x<\dfrac{\pi}{2}\\ a-8, & x=\dfrac{\pi}{2}\\ (1+|\cot x|)^{\frac{b}{a}|\tan x|}, & \dfrac{\pi}{2}<x<\pi\end{cases}$ where $a,b\in\mathbb{Z}$. If $f$ is continuous at $x=\dfrac{\pi}{2}$, then $a^2+b^2$ is equal to:

$\displaystyle\lim_{n\to\infty}\frac{(1^2-1)(n-1)+(2^2-2)(n-2)+\cdots+((n-1)^2-(n-1))\cdot1}{(1^3+2^3+\cdots+n^3)-(1^2+2^2+\cdots+n^2)}$ is equal to

997. 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.]

If $y = \tan^{-1}\!\!\sqrt{\dfrac{1+\sin x}{1-\sin x}}$, $x\in\!\left(0,\dfrac{\pi}{2}\right)$, then $\dfrac{dy}{dx}$:

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 R . Then \sum 100 n=1 log e f (n) is equal to :

882. Let $f(x) = \begin{cases} \dfrac{ax^3 + bx^2 + cx + d}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}$ be a continuous function where $a, b, c, d$ are in arithmetic progression. Then find the number of points where $|f(|x|)|$ is non derivable.

$\lim_{x \to \infty} \left(\frac{3x-4}{3x+2}\right)^{\frac{x+1}{3}} =$ ______

If $\lim_{x \to 2} \dfrac{\tan(x-2)\{x^2 + (k-2)x - 2k\}}{x^2 - 4x + 4} = 5$, then $k$ is equal to

A circular disk of unit radius is filled with a number of smaller circular disks arranged in the form of hexagon. Let $A_n$ denotes a stack of disks arranged in the shape of a hexagon having $n$ disks on a side. If $A$ be the area of large disk, $S_n$ be the number of disks in $A_n$ configuration and $r_n$ be the radius of each disk in $A_n$ configuration, then find $\lim_{n \to \infty} \frac{S_n}{n^2}$

If $y = \sum_{k=1}^{6} k\cos^{-1}\!\left\{\dfrac{3}{5}\cos kx - \dfrac{4}{5}\sin kx\right\}$, then $\dfrac{dy}{dx}$ at $x=0$ is: [Integer type]

If $y = (x^2+1)^{55}$, then $\dfrac{d^{55}}{dx^{55}}\!\big[(x^2+1)^{55}\big]_{x=1}$ equals $N\cdot55!$, find $N$. [Integer type; answer 2890]

If $\left(a+\sqrt{2}\,b\cos x\right)\!\left(a-\sqrt{2}\,b\cos y\right)=a^2-b^2$, where $a>b>0$, then $\dfrac{dx}{dy}$ at $\!\left(\dfrac{\pi}{4},\dfrac{\pi}{4}\right)$ is: [If answer expressed as $(a+b)/(a-b)$, find $(a+b)^2$ when $a=5,b=2$]

A circular disk of unit radius is filled with a number of smaller circular disks arranged in the form of hexagon. Let $A_n$ denotes a stack of disks arranged in the shape of a hexagon having $n$ disks on a side. If $A$ be the area of large disk, $S_n$ be the number of disks in $A_n$ configuration and $r_n$ be the radius of each disk in $A_n$ configuration, then find $\lim_{n \to \infty} n r_n$

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

Twice differentiable f. Find $(k_1,k_2)$.

Let $f(x) = \displaystyle\int_0^x t\ln(1+t^2)\, dt$, then $f''(0)$ is:

If $f(x)$ is continuous at $x = 0$ and $\lim_{x \to 0} f(x) = e^{\lim_{x \to 0^2} ab\left(\sin\left(\frac{2x^2}{a}\right) + \cos\left(\frac{3x}{b}\right) - 1\right)} = f(0) = e^3$, then the minimum value of $a$ for which $b$ is real is:

The limit limx→∞ (x+1)10+(x+2)10+···+(x+100)10 x10+1010 is:

Let f(x) = sin{x} x2+ax+b. If f(5+) and f(3+) exist finitely and are non-zero, find (a + b):

Let $f(x)$ be differentiable function on the interval $(0, \infty)$ such that $f(1) = 1$ and $\lim_{t \to x} \frac{t^3 f(x) - x^3 f(t)}{t^2 - x^2} = \frac{1}{2}$ for all $x > 0$, then $f(x)$ is:

The value of limx→0 tan π 4 + x 1/x is:

$f(x)=\begin{cases}a+\sin(\sin x) & x\ge 0\\ \ln(\cos x)+bx & x differentiable at \(x=0$. Then:

If $f(x) = \begin{cases} x+1; & x > 1 \\ 0; & x = 1 \text{ then } f'(0) \text{ equals to} \\ -3x; & x < 1 \end{cases}$

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

The value of $\lim_{x \to 0} \frac{\sin(2x + \tan 2x)}{x^3}$ is equal to

Let $[x]$ denote the greatest integer function and $f(x)=\max\{1+x+[x],\ 2+x,\ x+2[x]\}$, $0\leq x\leq 2$, where $m$ is the number of points where $f$ is not continuous and $n$ be the number of points in $(0,2)$ where $f$ is not differentiable. Then $(m+n)^2+2$ is equal to

Let x = 2 be a root of the equation x^2 + px + q = 0 and f(x) = \begin{cases} \dfrac{1-\cos(x^2-4px+q^2+8q+16)}{(x-2p)^4}, & x \neq 2p \\ 0, & x = 2p \end{cases}. Then \lim_{x \to 2p^+} [f(x)], where [.] denotes greatest integer function, is

If $\lim_{x \to \infty} \left(1 + p^2 + q^2(x)\right) = e^p$, then

If $f(x) = (x-1)(x-2)(x-3)(x-4)(x-5)$, then the value of $f'(5)$ is equal to

If limx→0[1 + x ln(1 + b2)]1/x = 2b sin2 θ, b > 0 and θ ∈(−π, π], then the value of θ is:

If limx→1 x4−1 x−1 = limx→k x3−k3 x2−k2 , then k is:

Same $f(x)=\text{tgn}(x)\cdot\sin(x)\cdot|x|$. Points of non-differentiability in $[-10,10]$:

If constants $a$, $b$ and $c$ so that $\lim_{x\to 0}\dfrac{axe^x - b\log(1+x) + cxe^{-x}}{x^2\sin x} = 2$ then find $(a+b-c)/8$.

$\displaystyle\lim_{x\to0}\left(\dfrac{1-\cos^2(3x)}{\cos^3(4x)}\right)\left(\dfrac{\sin^3(4x)}{(\ln(2x+1))^5}\right)$ is equal to

$f(x)=\begin{cases}e^{x^2}+x^3+1 & x>0\\ ax^2+bx+2 & x\le 0\end{cases}$ is differentiable at $x=0$. Then:

The value of limx→0 ln(sin 3x) ln(sin x) is:

The value of $\lim_{x \to 0} \frac{1-\cos(\tan x)}{x^2 \sin(2\sin(x)\cdot \cos(\frac{1}{x}))}$ is equal to

$f(x)=\begin{cases}\sin^{-1}x+a\cos\pi x & -1 is differentiable in \((-1,1)$. Find $2a+b$.

Define $f: [0, \pi] \to \mathbb{R}$ by $f(x) = \begin{cases} \tan^2 x + \sqrt{2\sin^2 x + 3\sin x + 4 - \sqrt{\sin^2 x + 6\sin x + 2}} & x \neq \pi/2 \\ k & x = \pi/2 \end{cases}$ is continuous at $x = \pi/2$, then $k$ is equal to:

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

If $x^2 + y^2$ and $z = x + 3t, y = 2x - t$, then $\frac{dz}{dy}$ is equal to (where $t$ is a constant)

Let f, g and h be the real valued functions defined on \mathbb{R} as f(x) = \begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{cases}, g(x) = \begin{cases} \frac{\sin(x+1)}{(x+1)}, & x \neq -1 \\ 1, & x = -1 \end{cases} and h(x) = 2[x] - f(x), where [x] is the greatest integer \leq x. Then the value of \lim_{x \to 1} g(h(x-1)) is

If $f(x) = \begin{cases} \frac{\sqrt{a+x}-\sqrt{a-x}}{x} & -1 \leq x < 0 \\ \frac{3x+2}{x-3} & 0 \leq x \leq 1 \end{cases}$ is continuous in $[-1, 1]$, then the value of $a$ is

If $f(x)$ is odd linear polynomial with $f(1) = 1$, then \[\lim_{x \to 0} \frac{2^{f(\tan x)} - 2^{f(\sin x)}}{x^2 f(\sin x)}\] is

Let $f(x)=\begin{cases}x^2\sin\left(\dfrac{1}{x}\right), & x\neq0\\0, & x=0\end{cases}$. Then at $x=0$:

Let f(x) = ( 1 + 2x a 0 ≤x

If $\displaystyle\lim_{x\to0}\frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}}{x\cos z-\log_e(1+z)}=2$, then $a^2+b^2+c^2$ is equal to:

Limits, Continuity & Differentiability Questions (1044)