Continuity practice · Mixed difficulty

The value of $k$ for which the function $f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{1 - \cos 4x}}{{8{x^2}}},x \ne 0}\\ {kx = 0} \end{array}} \right.$ is continuous at $a = 0$, is

The function
$f(x)= \begin{cases}\frac{1-\cos 4 x}{x^2} & \text { if } x < 0 \\ \frac{2 \sqrt{x}}{\sqrt{16+\sqrt{x}}-4} & \text { if } x=0 \\ \text { if } x > 0\end{cases}$
is continuous at $x=0$ for

Let $f\left( x \right)$ be a function differentiable at $x = c$. Then, $\lim\limits_{x \to c} f\left( x \right)$ equals

Let $f: R \rightarrow R$ be any function. Define $g: R \rightarrow R$ by $g(x)=|f(x)|$ for $x \rightarrow R$. Then, $g$ is

The function $f\left( x \right) = \frac{{1 - \sin x + \cos x}}{{1 + \sin x + \cos x}}$ is not defined at $x = \pi$. The value of $f\left( \pi \right)$, so that $f\left( x \right)$ is continuous at $x = \pi$, is

If $f: R \rightarrow R$ is a continuous function satisfying $f(0)=1$ and $f(2 x)-f(x)=x$ for all $x \in R$ and $\lim\limits _{x \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=P(x)$, then $P(x)$, is

If $f(x)=\left\{\begin{array}{cc}a x+1, & x \leq \pi / 2 \\ \sin x+b, & x>\pi / 2\end{array}\right.$ is continuous then

If $f(x)= \begin{cases}\cos ^{-1}(\cot x), & x < \frac{\pi}{2} \\ a(\pi[x]-1), & x \geq \frac{\pi}{2}\end{cases}$
The value of $a$ for $f$ to be continuous at $x=\frac{\pi}{2}$ is

The values of $x$ for which the tangents to the curves $y=x \cos x, y=(\sin x) / x$ are parallel to the axis of $x$ are roots of (respectively)

Statement 1: A function $f: R \rightarrow R$ is continuous at $x_{0}$ if and only if $\lim _{x \rightarrow x_{0}} f(x)$ exists and $\lim _{x \rightarrow x_{0}} f(x)=f\left(x_{0}\right)$

Statement 2: A function $f: R \rightarrow R$ is discontinuous at $x_{0}$ if and only if, $\lim _{x \rightarrow x_{0}} f(x)$ exists and $\lim _{x \rightarrow x_{0}} f(x) \neq f\left(x_{0}\right)$.

Given a function $f(x)$ continuous $\forall \, x \in R$ such that $\lim \limits_{{x \rightarrow 0}}\left[f(x)+\log \left(1-\frac{1}{e^{f(x)}}\right)-\log (f(x))\right]=0$, then $f(0)$ is

If $f: R \rightarrow R$ is defined by
$f(x) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{\cos 3x - \cos x}}{{{x^2}}},}&{x \ne 0}\\ {\lambda ,}&{x = 0} \end{array}} \right.$ and if $f$ is continuous at $x=0$, then $\lambda$ is equal to

If $f(x)$ is a continuous function satisfying $f(x) f(1 / x)=f(x)+f(1 / x)$ and $f(1)>0$ then $\lim \limits_{{x \rightarrow 1}} f(x)$ is equal to

The number of points at which the function $f\left( x \right) = \left( {\left| {x - 1} \right| + \left| {x - 2} \right| + \cos x} \right)$ where $x \in \left[ {0,4} \right]$ is not continuous, is

The function $f(x)=\cot x$ is discontinuous on the set

Let $f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {{x^P}\sin \frac{1}{x},}&{x \ne 0} \\ {0,}&{x = 0} \end{array}} \right.$ then $f(x)$ is continuous but not differentiable at $x = 0$ if

If $f:R \to R$ is defined by
$f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{2\sin x - \sin 2x}}{{2x\cos x}}}\\ {a,\mathrm{if}\,\,x = 0} \end{array}} \right.,\mathrm{if}\,\,x \ne 0,$
Then the value of a so that $f$ is continuous at 0 is

The function $f(x) = \lim\limits_{n \to \infty } \frac{{\cos (2\pi x) - {x^{2n}}\sin (x - 1)}}{{1 + {x^{2n + 1}} - {x^{2n}}}}$ is continuous for all $x$ in

If the tangent to the curve $x^3-y^2=0$ at $\left(m^2,-m^3\right)$ is parallel to $y=-\frac{1}{m} x-2 m^3$, then the value of $m^2$ is

Let $f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\int \limits_0^x \left\{ {5 + \left| {1 - t} \right|} \right\}dt,\mathrm{if}\,\,x > 2}\\ {5x + 1,\mathrm{if}\,\,x \le 2} \end{array}} \right.$

The function $f(x)=(x)$, where $(x)$ denotes the smallest integer $\geq x$, is

Let $f$ be a continuous function satisfying $f(x+y)=$ $f(x)+f(y)$ for all $x, y \in \mathbf{R}$ and $f(1)=5$ then $\lim \limits_{{x \rightarrow 4}}$ $f(x)$ is equal to

If $f(x)=\left\{\begin{array}{c}\frac{|x-a|}{x-a}, \text { when } x \neq a \\ 1, \text { when } x=a\end{array}\right.$, then

If the function
$f(x)=\left\{\begin{array}{cc} \frac{x^2-2 x+a}{\sin x} & \text { when } x \neq 0 \\ b & \text { when } x=0 \end{array}\right.$
is continuous at $x=0$, then

Let $f = \left( { - \frac{\pi }{4},\,\,\frac{\pi }{4}} \right) \to R$ be defined as
$f(x) = \left\{ {\begin{array}{cccccccccccccccccccc} {{{(1 + \left| {\sin \mathrm{ }x} \right|)}^{\frac{{3a}}{{\left| {\sin \mathrm{ }x} \right|}}}},}&{ - \frac{\pi }{4} < x < 0}\\ {b,}&{x = 0}\\ {{e^{\frac{{\cot \mathrm{ }4x}}{{\cot \mathrm{ }2x}}}},}&{0 < x < \frac{\pi }{4}} \end{array}} \right.$
If $f$ is continuous at $x = 0$, then the value of $6\,a + {b^2}$ is equal to

If $f(x)=\left\{\begin{array}{cl}\frac{((a-n) n x-\tan x) \sin n x}{x^2} & \text { at } x \neq 0, \\ 0 & \text { at } x=0\end{array}\right.$ where $n$ is a non-zero real number, and $f$ is continuous at $x=0$, then a is equal to

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

If $f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {mx + 1,x \le \frac{\pi }{2}}\\ {\sin x + n,x > \frac{\pi }{2}} \end{array}} \right.$ is continuous at $x = \frac{\pi }{2}$, then

Let $f:[1,3] \rightarrow R$ be a function satisfying $\frac{x}{[x]} \leq f(x) \leq \sqrt{6-x}$, for all $x \neq 2$ and $f(2)=1$, where $R$ is the set of all real numbers and $[x]$ denotes the largest integer less than or equal to $x$.

Statement 1: $\lim \limits_{x \rightarrow 2^{-}} f(x)$ exists.
Statement 2: $f$ is continuous at $x=2$.

If $f(x)=\left\{\begin{array}{cc}\frac{x}{e^{1 / x}+1}, & \text { when } x \neq 0 \\ 0 & \text {, when } x=0\end{array}\right.$, then