Differentiability Questions (1063)

For each x ∈R, let [x] be the GIF. Then limx→0−x([x]+|x|) sin[x] |x| is equal to: (1) −sin 1 (2) 0 (3) 1 (4) sin 1

limn→∞ en (1+ 1 n) n2 equals:

The value of the limit ℓis:

Let \(f(x) = \frac{\log_e(1+ax) - \log_e(1-bx)}{x}, x \neq 0\). The value to be assigned to \(f(x)\) at \(x = 0\) so that \(f(x)\) is continuous at \(x = 0\) is:

If \(f(x) = \log_e x\) then the differential coefficient of \(f(\log_e x)\) with respect to \(x\) is

Let \(a_n = \lim_{n \to 0}\left(\dfrac{a_{n-1}}{n}\right)^2 (f(h) - f(0))^2\) where \(f'(0) = 1\). If \(a_1 = 1\), find \(\prod_{i=1}^{10} a_i\).

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

Let Un = n! (n+2)! where n ∈N. If Sn = Pn r=1 Ur, then limn→∞Sn equals:

The value of d/dx tan2(x/(1-tan2x))cot 3x is

The value of \(\lim_{x \to 0}(\sin x)^x\) is

If f(x) = 1 + cos2(x2), then the value of f′(π/6) is

If y = sin x + y, then dy/dx is equal to

The value of limx→∞ √ x2 −2x −1 − √ x2 −7x + 3  is:

Let L = limx→0 a− √ a2−x2−x2 4 x4 , a > 0. If L is finite, then:

Find the value of \(\lim_{x \to 0} \frac{\sin^2 x}{2 - \sqrt{1 + \cos x}}\)

If \(\lim_{x \to 0} \left[1 + x + \frac{f(x)}{x}\right]^{1/x} = e^3\), then \(\lim_{x \to 0} \frac{f(x)}{x}\) is equal to

If a is the positive root of the equation p(x) = x2 − x − 2 = 0, then limx → a 1 − cos(πp(x))/x2 + a − 4 is equal to

The value of \(\lim_{x \to \pi/4} (1 + [x])^{\log(\tan x)}\) (where \([\cdot]\) denotes greatest integer function) is

If \(\lim_{x \to 0} \frac{1}{x^8}\left[\cos\frac{x^2}{2} - \cos\frac{x^2}{4} + \cos\frac{x^2}{2} - \cos\frac{x^2}{4}\right] = 2k\), then the value of \(k\) is ……… .

Find \(\lim_{x \to 0} \frac{\log \log (1 - x^2)}{\log \log \cos x}\)

If f(x) = cos x × cos 2x × cos 4x × cos 8x × cos 16x, then f'(π/4) is

If y = sin⁻¹((1-x)/(1+x)), 0 , then dy/dx is

The points of discontinuity of \(y = \frac{1}{u^2 + u - 2}\), where \(u = \frac{1}{x-1}\), are:

Given \(f(0) = 0\) and \(f(x) = \frac{1}{1-e^{-1/x}}\) for \(x \neq 0\). Then, only one of the following statements on \(f(x)\) is true.

If x\log_e(\log_e x) - x^2 + y^2 = 4 (y > 0), then \frac{dy}{dx} at x = e is equal to

If \(f(x) = \cos\left[\frac{\pi}{x}\right]\cos\left(\frac{\pi(x-1)}{2}\right)\), where \([\cdot]\) denotes the greatest integer function, then \(f(x)\) is continuous at

Let \((\tan\alpha)x + (\sin\alpha)y = \alpha\) and \((\alpha\operatorname{cosec}\alpha)x + \cos\alpha y = 1\) be two variable straight lines, \(\alpha\) being the parameter. Let P be the point of intersection of the lines. If the coordinates of P in the limiting position when \(\alpha \to 0\) be \((h, k)\) then

Given,\[f(x) = \begin{cases} \frac{\cos 2x - \sin 2x - 1}{x^2 + x - 2}, & x \neq \text{?} \\ a, & x = \text{?} \end{cases}\]where f is continuous at the point where the function is defined piecewise. Find the constant a.

If f'(x) = g(x) and g'(x) = -f(x) for all x and f(2) = 4 = f'(2), then (f(24))^2 + (g(24))^2 is

If the function f(x) = {a|p - x| + 1, x ≤ 5; b|x - p| + 3, x > 5} is continuous at x = 5, then the value of a - b is

If \(y = \tan^{-1}(\sec x - \tan x)\), then \(\frac{dy}{dx}\) is equal to

If $\displaystyle \lim_{x \to a} \frac{f(x) - f(a)}{(x-a)^3}$ is a finite non-zero number, then $f(x)$ is of maximum degree

The value of $\lim_{x \to 0} \sin^{-1}\{x\}$ (where $\{\cdot\}$ denotes fractional part of $x$) is

\(\lim_{x \to \infty} C_n^m x^m \left(1 + \frac{1}{n}\right)^{nx} \left(1 + \frac{1}{x}\right)^m\) equals to

Let \(h(x) = \min\{x; x^2\}\) for every real number \(x\). Then, which one of the following is true?

The value of $\lim_{x \to 0} \frac{\tan(\{1/x\}) \sin(ax)}{\sin(bx)}$, where $\{x\}$ denotes the fractional part function is

If f(x) = \begin{cases} A \sin[x] & \text{for } [x] \geq 0 \\ 0 & \text{for } [x] , where $[x]$ denotes the greatest integer less than or equal to $x$, then $\lim_{x \to 0} f(x)$ equals

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: