Limits, Continuity & Differentiability Questions (1044)

If limx→0 x2 sin βx αx−sin x = 1, find 6(α + β).

Let $y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $\dfrac{d}{dx}\!\big[y(x)\big]_{x=1}$ equals: [Integer type]

Let \(|f(x)-f(y)|\le 2|x-y|^{3/2}\), \(f(0)=1\). Find \(\int_0^1 f^2(x)\,dx\).

Given \(f\) piecewise defined. \(f_1(x)=|f(|x|)|\). If \(f\) is continuous in \([-2,10]\), then:

The value of limx→0 | cos(sin 3x)|−1 x2 is:

Evaluate: \(\lim_{x \to 0} \left(\tan \frac{x}{x}\right)^{\frac{1}{x}}\)

Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in A.P., then $f'(a)$, $f'(b)$, $f'(c)$ are in:

Suppose \(f\) is a derivable function that satisfies the equation \( f(x+y) = f(x) + f(y) + x^2y + xy^2 \) for all real numbers \(x\) and \(y\). Suppose that \( \lim_{x \to 0}\left[\dfrac{f(x)}{x}\right] = 1 \), find \( f(3) = \) __________.

The value of $\lim_{x \to 0} \frac{(\tan x - \sin x)}{(\tan x - \sin x) + (\tan x + \sin x)} + \frac{1}{r^3 - r}$ is

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.

The value of \(\lim_{n \to \infty} \sqrt[n]{a^n + b^n}\) (where \(a > 0, b > 0\)) is equal to

If $y = e^{\sin^{-1}x} + e^{\cos^{-1}x}$, then which of the following is/are true?

Let fn(x) = limt→x \(f_n(x) = \lim_{t \to x} \frac{\sin^{-1}(nt)}{2t} = \frac{\sin^{-1}(nx)}{2x}\). Find the value of \(\lim_{x \to 0} \left[\frac{\sin^2 2x}{2x}\right] + \left[\frac{\sin^{-1} 4x}{2x}\right]\).

279. Let \( f(x) = \dfrac{x \ln x - \ln x}{9x^2 - 2e^x x - 9x + 2e^x} + 2 \) and \( g(x) = \sin^2\left(\dfrac{\pi x^2}{2}\right) \), then the value of \( \lim_{x \to 1} \dfrac{f(x)}{g(x)} \) is:

For the curve \sin x + \sin y = 1 lying in the first quadrant, there exists a constant a for which \lim_{x \to 0} x^a \frac{d^2y}{dx^2} = L (not zero). Find 2a.

Evaluate: \(\lim_{x \to \infty} \left(\frac{3x^2 + 1}{4x^2 - 1}\right)^{\frac{x^2}{1+x}}\)

Let \( f(x) = \cot^{-1}\left(\text{sgn}\left(\dfrac{[x]}{2x - [x]}\right)\right) \):Statement-1: \( f(x) \) is discontinuous at \( x = 1 \).Statement-2: \( f(x) \) is non-differentiable at \( x = 1 \).Which of the following option is correct?[Note: \([k]\) denotes greatest integer function less than or equal to \(k\).]

Let a = min{x2 + 2x + 3, x ∈R} and b = limθ→0 1−cos θ θ2 . Then the value of Pn r=0 arbn−r is:

If $x = \sec\theta - \cos\theta$ and $y = \sec^n\theta - \cos^n\theta$, then $\left(x^2+4\right)\!\left(\dfrac{dy}{dx}\right)^{\!2}$ equals $n^2(y^2+k)$. Find $k$.

If $f(x) = \frac{2\sin x \cos x + 2x \cot x}{2\sin x}$, find $f'\left(\frac{\pi}{4}\right)$ and verify it equals $\sqrt{2}$

Let \(f(x) = \sin^2(\sin x)\). If \(g\) is the inverse of \(f\), find \(g''(3)\).Given: \(f'(0) = \sin^2(\sin 1)\), \(f''(0) = 2\sin(\sin 1)\cos(\sin 1)\cos 1\)\[g''(y) = \frac{-1}{[f'(x)]^3} f''(x)\]

Let $y = \cot^{-1}(1) + \cot^{-1}(2) + \cot^{-1}(3) + \cdots$. If $\dfrac{d}{dn}\!\left[\sum_{r=1}^{n}\cot^{-1}(r)\right]$ is evaluated and the sum $\sum_{r=1}^{10}\cot^{-1}(r^2-r+1)$ equals $\tan^{-1}(k/l)$, find $10k+l$ (answer 101 from key).

If $y^2 + \ln(\cos^2 x) = y$, then $|y''(0)+y'(0)|$ equals: [Integer type]

If \(f(x) = \begin{cases} \frac{\tan^2\{x\}}{x}, & x > 0 \\ 1, & x = 0 \\ \{x\}\cot\{x\}, & x where \([x]\) is the integral part of \(x\) and \(\{x\}\) is the fractional part of \(x\), then

Let $f:(−\infty,\infty)−\{0\}\to\mathbb{R}$ be a differentiable function such that $f'(1)=\displaystyle\lim_{a\to\infty}a^2f\!\left(\frac{1}{a}\right)$. Then $\displaystyle\lim_{a\to\infty}\frac{a(a+1)}{2}\tan^{-1}\!\left(\frac{1}{a}\right)+a^2-2\log_e a$ is equal to

For each \( x \in \mathbb{R} \), let \( [x] \) be the greatest integer less than or equal to x. Then \( \lim_{x \to 0} \frac{x[|x|] \sin[x]}{|x|} \) is equal to

For each \( t \in \mathbb{R} \), let \( [t] \) be the greatest integer less than or equal to t. Then \( \lim_{x \to 1^-} \frac{(1-|x|+\sin[1-x])\sin\left(\frac{\pi}{2}[1-x]\right)}{|1-x|[|1-x|]} \)

If \(\lim_{x \to -1} \frac{x^2 - ax + b}{x - 1} = 5\), then \(a + b\) is equal to :-

\[ \lim_{x \to 2} \frac{\sqrt{1 - \cos{2(x - 2)}}}{x - 2} \]

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

Given: \(x^2 + y^2 + \sin y = 4\). Find \(-\dfrac{d^2y}{dx^2}\bigg|_{(-2,0)}\).

If $f(x) = \log_x(\log_e x)$, then $f'(e)$ equals:

If \(f(2) = 2\), \(f'(2) = 1\) then \(\lim_{x \to 2} \frac{2x^2 - 4f(x)}{x - 2} =\) ______

Given \(f'(3) + f'(2) = 0\). Let \[y = \lim_{x \to 0} \left[\frac{1 + f(3+x) - f(3)}{1 + f(2-x) - f(2)}\right]^{1/x}\] Find the value of \(y\).

If α = lim x→0+ e √ tan x −e √x √ tan x −√x and β = lim x→0(1 + sin x) 1 2 cot x are the roots of the quadratic equation ax2 + bx −√e = 0, then 12 loge(a + b) is equal to

\(f(x)=\begin{cases}1-x & 0\le x\le 1\\ x+2 & 1. For \(y=f(f(x))\) on \([0,4]\):

Let \(g(x)=\begin{cases}3x^2-4\sqrt{x}+1 & x. If \(g(x)\) is continuous and differentiable at \(x=1\), find \(a\) and \(b\).

Let f : (−1, 1) → ℝ be continuous and \[\int_0^{\sin x} f(t)\,dt = \frac{\sqrt{3}}{2}\,x.\] Find \(f\!\left(\dfrac{\sqrt{3}}{2}\right)\).

The set of all values of a for which \lim_{x \to a}\left([x-5] - [2x+2]\right) = 0, where [\alpha] denotes the greatest integer less than or equal to \alpha, is equal to

Let \(f\) be a function defined by \(y = f(x)\) where \(x = 2t - |t|\) and \(y = t^2 + t|t|\) for \(t \in \mathbb{R}\), then:

If \(y = e^{nx}\) then \(\left(\dfrac{d^2y}{dx^2}\right)\left(\dfrac{d^2x}{dy^2}\right)\) is equal to

Given: \(2y = \left(\cot^{-1}\left(\dfrac{\sqrt{3}\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)\right)^2\). Find \(\dfrac{dy}{dx}\) (or simplify \(2y\)).

If \(f(x)\) and \(g(x)\) are not differentiable finitely at a point then will \(f(x) \cdot g(x)\) will also be nondifferentiable finitely at that point?

Given: \(x = \sqrt{2^{\cosec^{-1}t}}\) and \(y = \sqrt{2^{\sec^{-1}t}}\) where \(|t| \geq 1\). Find \(\dfrac{dy}{dx}\).

Let \(f(x)\) be a continuous and differentiable function such that \(\displaystyle\lim_{h \to 0} \frac{f(3+7h) - f(3+4h)}{h} = 4\). Then the value of \(f'(3)\) equals:

\(\lim_{x \to 2} \dfrac{\sqrt{2\sin^2(x-2)}}{x-2}\)

$\dfrac{d}{dx}\left[\tan^{-1}\\!\left(\dfrac{\sqrt{2-x}}{1+x^2}\right)\right]$ equals $(x \ge 0)$:

Find \(a\), \(b\) and \(c\) such that \[\lim_{x \to 0} \frac{ax e^x - b\log(1+x) + cx e^{-x}}{x^2 \sin x} = 2\]

Let $f(x) = x + \sin x$. Suppose $g$ denotes the inverse function of $f$. The value of $g'\\!\left(\frac{\pi}{4} + \frac{1}{\sqrt{2}}\right)$ has the value equal to:

If $y = x + e^x$, then $\dfrac{d^2x}{dy^2}$ is: