Limits, Continuity & Differentiability Questions (1044)

If the function \(g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3

If the function \[f(x) = \begin{cases} -x, & x

If \(f(x) = |x|\) then \(f'(x) = \frac{|x|}{x}, x \neq 0\).State whether this statement is true or false.

The value of \(k\) for which the function \[f(x) = \begin{cases} \left(\dfrac{4}{5}\right)^{\frac{\tan 4x}{\tan 5x}}, & 0

Let \( f(x) \) be continuous and differentiable everywhere and defined as \[ f(x) = \begin{cases} ax^2 - b & x

If $\Delta_1 = \begin{vmatrix} x & b & b \\ a & x & b \\ a & a & x \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} x & b \\ a & x \end{vmatrix}$, then $\dfrac{d}{dx}(\Delta_1) = 3(\Delta_2)^{1/2}$... find the value (A: $3(\Delta_2)^{1/2}$):

Let $f(x) = \sin\!\left(\cos^{-1}(1-x^2)\right) + \cos\!\left(\sin^{-1}(2x-2x^3)\right)$. For $x\in(0,1/\sqrt{2})$, $f'(x)$ equals:

Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f'(2)=5$, $f'(x)\geq 1$ and $f''(x)\geq 4$ for all $x\in(1,6)$, then:

If f(x) = \log_2(\log x), then f'(x) at x = e is

Let f be a polynomial of degree 2018 such that f(1) = 1, f(2) = 0, f(3) = −5, f(−4) = 2. If f(x) is an even function then find the minimum number of points where f″(x) = 0.

\(\lim_{x \to 0} \frac{\sin\left(\pi \cos^2(\tan(\sin x))\right)}{x^2} = \)

Consider the function \( f(x) = |x^2 - 7x + 12|(x^2 - 7x + 10)(x^2 - 4x + 3) \). Then Rolle's theorem for \( f(x) \) is not applicable to which of the following range?

$\sin ax + \cos ax$ and $|\cos x| + |\sin x|$ are periodic functions of same fundamental period, if '$a$' equals

limx→0  3x2+2 7x2+2 1/x2 is equal to:

237. 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:

The number of points in (1, 3), where \(f(x) = a[x^2]\), \(a > 1\), is not differentiable, where [x] denotes the integral part of x.

The value of limx→1 3√ 7+x3− √ 3+x2 x−1 is:

Evaluate: \[\lim_{x \to \frac{\pi}{2}} \frac{\cot x(1 - \sin x)}{-8\left(x - \dfrac{\pi}{2}\right)^3}\]

\(\lim_{x \to \pi/2} \dfrac{\left[1 - \tan\left(\dfrac{x}{2}\right)\right][1 - \sin x]}{\left[1 + \tan\left(\dfrac{x}{2}\right)\right][\pi - 2x]^3}\) is

limx→0 tan π 4 + x

1/x is equal to:

limx→∞  11/x+21/x+31/x+···+n1/x n nx where n ∈N is equal to:

Let f : R → R be a continuous function satisfying \( f(x) + \int_0^x t f(t)\, dt + x^2 = 0 \) ∀ x. Then:

Differentiable on ℝ. Find \(48(a+b)\).

lim x→1 (5x + 1)1/3 −(x + 5)1/3 (2x + 3)1/2 −(x + 4)1/2 = m √ 5 n(2n)2/3 , where gcd(m, n) = 1. Then 8m + 12n is equal to

limx→∞ (√3x+1+√3x−1)6+(√3x+1−√3x−1)6 (x+ √ x2−1)6+(x− √ x2−1)6 · x3

If limx→0 sin x+aex+be−x+c ln(1+x) x3 is finite, then a −b + c is:

Find f(x) = limn→∞(cos x √n)n.

If $f(x) = \begin{cases} \frac{[(a-n)nx - \tan x]\sin nx}{x^2} & \text{at } x = 0 \\ 0 & \text{at } x = 0 \end{cases}$, where $n$ is a non-zero real number, and $f$ is continuous at $x = 0$, then $a$ is equal to:

Let $f$ be a function defined on $(-\pi/2, \pi/2)$ as follows: $f(x) = \begin{cases} \frac{2^{[1/n]} - [x] - \frac{[x]}{[n2-1]}}{x\tan x} & x \neq 0 \\ k & x = 0 \end{cases}$. The value of $k$ so that $f$ is continuous at $x = 0$ is:

If \(f(x)=x(\sqrt{x}-\sqrt{x+1})\), then:

If \(f(x)=\text{sgn}(x^5)\), which are false?

Let \(S\) be the set of all points where \(\sqrt[5]{x^2|x|^3}-\sqrt[3]{x^2|x|}-1\) is not differentiable. \(S\) is a subset of:

Number of points where \(f(x)=|x-\text{sgn}(x)|\) is non-differentiable (sgn(.) denotes signum function):

Let \(f(x)=\begin{cases}x^3+2x^2 & x\in\mathbb{Q}\\ -x^3+2x^2+ax & x\notin\mathbb{Q}\end{cases}\). Integral value of \(a\) so that \(f\) is differentiable at \(x=1\):

If $f(x) = |\cos x - \sin x|$, then $f'\!\left(\dfrac{\pi}{6}\right)$ equals (give answer as integer after multiplying by $-1$ if negative):

If $y = \sqrt{\dfrac{1-x}{1+x}}$, then $\dfrac{dy}{dx}$ equals:

\(f(x)=\begin{cases}\frac{\ln\cos x}{ax} & x>0\\ 0 & x=0\\ \frac{e^{x^2}-1}{bx} & x. If \(f'(0)=\frac{1}{4}\), then:

Which function is continuous everywhere in its domain but has at least one point where it is not differentiable?

Given function \(f(x) = x^3 + x^2 f'(1) + xf''(2) + f'''(3)\). Then \(f(2)\) equals:

Let \(f: R \to R\) be differentiable at \(c \in R\) and \(f(c) = 0\). If \(g(x) = |f(x)|\), then at \(x = c\), \(g\) is

\lim_{x \to \infty} \frac{\left(\sqrt{3x+1}+\sqrt{3x-1}\right)^6 + \left(\sqrt{3x+1}-\sqrt{3x-1}\right)^6}{\left(x+\sqrt{x^2-1}\right)^6 + \left(x-\sqrt{x^2-1}\right)^6} \cdot x^3

Given equation is $\ln(x+y) = 2xy$. Find $\frac{dy}{dx}$ at point $(0,1)$.

Given, $y = z^x$, then $y = z^y$. Find $\frac{dy}{dx}$.

\[\lim_{x \to 0} \frac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}\] is equal to: (where \([\,]\) denotes greatest integer function)

\(\lim_{x \to 0} \frac{f(x) \cdot g(x)}{x(1-g(x))}\) will be

If $f(x) = \cot^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)$ and $g(x) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$, then $$\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}, \quad 0

If \(x + |y| = 2y\) then \(y\) as a function of \(x\) is

197. Let \(g(x) = \dfrac{1}{f^{-1}(x)}\). Given the following data:\(x\)01234\(f(x)\)\(-2\)\(-1\)246\(f'(x)\)1/22/314/35/3The value of \(g'(4)\) is:

If $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x\end{vmatrix}$, find $f'(1)$.

Let \(f: R \to R\) be a function defined as \[f(x) = \begin{cases} 5, & \text{if } x \leq 1 \\ a + bx, & \text{if } 1