Differentiability Questions (1063)

$\lim_{x \to \infty} \dfrac{(2x^2 - 3x + 5)(3x-1)^{x/2}}{(3x^2 + 5x + 4)\sqrt{(3x+2)^x}}$ is equal to:

\(f\) triangle function, \(g=f(x+2)-f(x-2)\). \(n+m\)?

If the function $f(x)=\begin{cases}\dfrac{1}{|x|} & ,\; |x|\ge 2\\ax^2+2b & ,\; |x|<2\end{cases}$ is differentiable on $\mathbb{R}$, then $48(a+b)$ is equal to ______.

The value of \(\displaystyle\lim_{x\to\frac{\pi}{2}}\frac{4(x-\pi)\cos^2 x}{\pi(\pi-2x)\tan\!\left(x-\dfrac{\pi}{2}\right)}\) is equal to:

Let \(f(x) = \begin{cases} \left[1 + \ln(c^2 + c + 1)\tan^2(x-1)\right]^{\frac{1}{(\ln x)^2}}, & x \neq 1 \\ 3c, & x = 1 \end{cases}\) where \(c \in R\). If \(\lim_{x \to 1} f(x)\) exists but \(f(x)\) is discontinuous at \(x = 1\), then \(c\) can take the value:

201. Let \(f: \mathbb{R} \to \mathbb{R}\) be a function such that for all \(x, y \in \mathbb{R}\), \(|f(x) - f(y)| \leq 6|x - y|^2\). If \(f(3) = 6\), then \(f(6)\) is equal to:

Consider the function $f:(0,2)\to\mathbb{R}$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $$g(x)=\begin{cases}\min\{f(t)\},\; 0<t\le x & ,\; 0<x\le 1\\\frac{3}{2}+x & ,\; 1<x<2\end{cases}$$ Then

The set of points where \( f(x) = \dfrac{x}{1+|x|} \) is differentiable, is

Let \(f: R \to R\) defined by \(f(x) = x^3 + 3x + 1\) and \(g\) be the inverse of \(f\), then the value of \(g''(5)\) equals:

Same \(f\) and \(f_1(x)=|f(|x|)|\). Number of points in \([-2,10]\) where \(f_1\) is NOT differentiable:

\(f=\max\{1+x+[x],2+x,x+2[x]\}\) on \([0,2]\). \((m+n)^2+2=\)

f piecewise. m+n = non-cont + non-diff in (-2,2):

200. If \(y = 2\tan^{-1}\!\left(\dfrac{\sqrt{1+x^2}-1}{x}\right)\), then the value of \(\dfrac{d^2y}{dx^2}\) at \(x = 2\) is:

Let \(f(x) = x|x|\) and \(g(x) = \sin x\). Then \(g \circ f\) is:

Let $\displaystyle\lim_{n\to\infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2n}{(n^2+1)\sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8n}{(n^2+4)\sqrt{n^4+16}}+\cdots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2n\cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}}\right)$ be $\dfrac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $k^2$ is equal to

If \(x^n = x^2 + x + 1\), then \(\lim_{n \to \infty} x^n\) (as \(n \to \infty\), \(x \to 1\)) equals:

Let a be a positive real number. Let f : R →R and g : (a, ∞) →R be defined by f(x) = sin πx 12  , g(x) = 2 loge(√x −√a) loge(e √x −e √a) . Then the value of limx→a+ f(g(x)) is

If \(y^2 = P(x)\) which is a polynomial of degree 3, then \(2\dfrac{d}{dx}\left(y^3 \dfrac{d^2y}{dx^2}\right)\) equals

If limx→0 eax−cos(bx)−cxe−cx 2 1−cos 2x = 17, find 5a2 + b2.

\(\lim_{x \to 0} \frac{\sqrt{\frac{1}{2}(1-\cos 2x)}}{x}\) is equal to

Let f(x) = ( sin x x ∈Z 0 x /∈Z and g(x) =      x2 + 1 x ̸= 0, 2 4 x = 0 5 x = 2 , then:

limx→1− √π− √ 2 sin−1 x √1−x is equal to:

Given \((2x)^{2y} = 4e^{2x-2y}\), then \(\dfrac{dy}{dx}(1 + \log_e 2x)^2\) equals:

Limit check for piecewise f(x):

Let \( S = \{t \in R : f(x) = |x - \pi| \cdot (e^{|x|} - 1)\sin|x| \) is not differentiable at \(t\}\). Then the set S is equal to:

If limx→1 x+x2+···+xn−n x−1 = 820, then n is equal to:

Find g(x) = −x4b where b = limx→∞( √ x2 + x + 1 − √ x2 + 1).

If limx→1 x2−ax+b x−1 = 5, then a + b is equal to:

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

Number of solutions of f(x) + g(x) = 0?

limx→2 3x+33−x−12 3−x/2−31−x is equal to:

limx→0 sin2(π cos4 x) x4 is equal to:

If \(f(0) = 1\) and \(\displaystyle\lim_{t \to x} \frac{\sec x \cdot f(t) - f(x) \sec t}{t - 1} = \sec^2 x\). The value of \(\dfrac{f(0)}{f'(0)}\), is:

The value of \(\displaystyle\lim_{x \to 0^+} \dfrac{\displaystyle\int_0^{\arctan x} (\sin t^2)\, dt}{x \cos x - x}\) is equal to:

\(\lim_{x \to 1^-} \dfrac{\sqrt{\pi} - \sqrt{2\sin^{-1}x}}{\sqrt{1-x}}\) is equal to __________ (up to four decimal places).

Let [x] denote the greatest integer function. Find limx→0 tan(π sin2 x)+(|x|−sin |x|)[x]2 x2 :

If \( f \) is a real-valued differentiable function satisfying \( |f(x) - f(y)| \leq (x-y)^2 \), \( x, y \in R \) and \( f(0) = 0 \), then \( f(1) \) equals

Let \(f: R \to R\) defined by \(f(x) = x^3 + 3x + 1\) and \(g\) be the inverse of \(f\), then the value of \(g''(5)\) equals:

\(\lim_{h \to 0} \frac{2\left[\sqrt{3}\sin\left(\frac{\pi}{6}+h\right) - \cos\left(\frac{\pi}{6}+h\right)\right]}{\sqrt{3}h(\sqrt{3}\cos h - \sin h)}\) is equal to

If \(f(x) = \frac{\sin[x]}{[x]}, [x] \neq 0\)\(= 0, [x] = 0\)where \([x]\) is the greatest integer function, then \(\lim_{x \to 0} f(x)\) equals:

If \(f(x) = \int_0^x |t| dt\), then

limx→π/2 tan2 x[ p 2 sin2 x + 3 sin x + 4 − p sin2 x + 6 sin x + 2]

If f(x) is an odd linear polynomial with f(1) = 1, then limx→0 2f(tan x)−2f(sin x) x2f(sin x) is:

Let f(x) = tan x x , then the value of limx→0([f(x)] + x2)1/{f(x)} is:

limx→1+ (1−|x|+sin |1−x|) sin( π 2 [1−x]) |1−x|[1−x] is equal to: (1) −1 (2) 1 (3) D.N.E (4) 0

\( f(x) = [\log_e x] + \sqrt{\{\log_e x\}}, x > 1 \), where [.] and {.} denote the greatest integer function and the fractional part function respectively, then

Properties of f(x) = limn→∞ (tan x)2n+x2 sin2 x+(tan x)2n for x ∈(−π/2, π/2):

Assume that \(f\) is continuous on \([a, b]\), \(a > 0\) and differentiable on \((a, b)\). If \(\dfrac{f(a)}{a} = \dfrac{f(b)}{b}\), then there exists \(x_0 \in (a, b)\) such that:

limn→∞ 1+2−3+···+(3n−2)+(3n−1)−3n √ 2n4+4n+3− √ n4+5n+4

If f(x) = cos 2−cos 2x x2−|x| , then: