Limits Questions (1092)

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

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: