Limits, Continuity and Differentiability

Let a function $f:\mathbf R\to\mathbf R$ be defined as :

$$f(x)=\begin{cases}\displaystyle\int_0^x(5-|t-3|)\,dt, & x>4\\ x^2+bx, & x\le4\end{cases}$$

where $b\in\mathbf R$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

If for $p\ne q\ne0$, the function $f(x)=\dfrac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then :

Let $f(x)=2+|x|-|x-1|+|x+1|$, $x\in\mathbf R$.

Consider

(S1) : $f'\left(-\dfrac32\right)+f'\left(-\dfrac12\right)+f'\left(\dfrac12\right)+f'\left(\dfrac32\right)=2$

(S2) : $\displaystyle\int_{-2}^2f(x)\,dx=12$

Then,

$\displaystyle\lim_{x\to0}\left(\frac{(x+2\cos x)^3+2(x+2\cos x)^2+3\sin(x+2\cos x)}{(x+2)^3+2(x+2)^2+3\sin(x+2)}\right)^{\frac{100}x}$ is equal to ______.

The function $f:\mathbf R\to\mathbf R$ defined by $f(x)=\displaystyle\lim_{n\to\infty}\frac{\cos(2\pi x)-x^{2n}\sin(x-1)}{1+x^{2n+1}-x^{2n}}$ is continuous for all $x$ in :

If $\displaystyle\lim_{x\to0}\frac{\alpha e^x+\beta e^{-x}+\gamma\sin x}{x\sin^2x}=\frac23$, where α, β, γ ∈ R, then which of the following is NOT correct ?

Let the solution curve $y=y(x)$ of the differential equation $\left(1+e^{2x}\right)\left(\dfrac{dy}{dx}+y\right)=1$ pass through the point $\left(0,\dfrac\pi2\right)$. Then, $\displaystyle\lim_{x\to\infty}e^xy(x)$ is equal to :

The number of points, where the function $f:\mathbf R\to\mathbf R$, $f(x)=|x-1|\cos|x-2|\sin|x-1|+(x-3)|x^2-5x+4|$, is NOT differentiable, is :

Let the function $f(x)=\begin{cases}\dfrac{\log_e(1+5x)-\log_e(1+\alpha x)}x & ;\ \text{if } x\ne0\\ 10 & ;\ \text{if } x=0\end{cases}$ be continuous at x = 0. Then α is equal to

If [t] denotes the greatest integer ≤ t, then the number of points, at which the function $f(x)=4|2x+3|+9\left[x+\dfrac12\right]-12[x+20]$ is not differentiable in the open interval (−20, 20), is _______.

Let

$$f(x)=\begin{cases}\dfrac{\sin(ax+3x)+\sin x}{x}, & x<0\\ 2, & x=0\\ \dfrac{(x+6bx^2)^{1/3}-x^{1/3}}{3x^{4/3}}, & x>0\end{cases}$$

be continuous at x = 0. If $(1-k)\,f\left(\dfrac7{18}\right)=ab$, then k is equal to

For any positive integer $n$, let $S_n: (0, \infty) \to \mathbb R$ be defined by
$$S_n(x) = \sum_{k=1}^n \cot^{-1}\left(\frac{1 + k(k+1)x^2}{x}\right),$$
where for any $x \in \mathbb R$, $\cot^{-1}(x) \in (0, \pi)$ and $\tan^{-1}(x) \in \left(-\frac\pi2, \frac\pi2\right)$. Then which of the following statements is (are) TRUE?

Let $f: \left[-\frac\pi2, \frac\pi2\right] \to \mathbb R$ be a continuous function such that
$$f(0) = 1 \quad\text{and}\quad \int_0^{\pi/3} f(t)\,dt = 0$$
Then which of the following statements is (are) TRUE?

$\displaystyle\lim_{x\to0}\frac{\int_0^{x^2}\left(\sin\sqrt t\right)dt}{x^3}$ is equal to :

If $f:\mathbf R\to\mathbf R$ is a function defined by $f(x)=[x-1]\cos\left(\dfrac{2x-1}2\right)\pi$, where [·] denotes the greatest integer function, then f is :

Let the function $f:\mathbb R\to\mathbb R$ be defined by $f(x) = x^3 - x^2 + (x-1)\sin x$ and let $g:\mathbb R\to\mathbb R$ be an arbitrary function. Let $fg:\mathbb R\to\mathbb R$ be the product function defined by $(fg)(x) = f(x)g(x)$. Then which of the following statements is/are TRUE?

Let $e$ denote the base of the natural logarithm. The value of the real number $a$ for which the right hand limit

$$\lim_{x\to0^+}\frac{(1-x)^{\frac1x} - e^{-1}}{x^a}$$

is equal to a nonzero real number, is ______

Let the functions $f:(-1,1)\to\mathbb R$ and $g:(-1,1)\to(-1,1)$ be defined by

$$f(x) = |2x-1| + |2x+1|\quad\text{and}\quad g(x) = x - [x],$$

where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f\circ g:(-1,1)\to\mathbb R$ be the composite function defined by $(f\circ g)(x) = f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f\circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f\circ g$ is NOT differentiable. Then the value of $c + d$ is ______

The value of the limit

$$\lim_{x\to\frac\pi2}\frac{4\sqrt2(\sin3x + \sin x)}{\left(2\sin2x\sin\frac{3x}{2} + \cos\frac{5x}{2}\right) - \left(\sqrt2 + \sqrt2\cos2x + \cos\frac{3x}{2}\right)}$$

is ______

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be functions satisfying

$$f(x+y) = f(x) + f(y) + f(x)f(y)\quad\text{and}\quad f(x) = xg(x)$$

for all $x, y\in\mathbb R$. If $\lim_{x\to0} g(x) = 1$, then which of the following statements is/are TRUE?