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Recent questions and answers in Limit, Continuity and Differentiability

Questions >> JEEMAIN and NEET >> Mathematics >> Limit, Continuity and Differentiability

$\lim\limits_{x \to 2} \frac{tan(x-2) \left | x^2+(k-2)x-xk \right | }{(x-2)^2}=5$ ..find k

answered Dec 28, 2018 by ravindrayadav78795

1 answer

If $f(x)=(x+1)^{\cot x}$ is continuous at $x=0$ then $f(0)$ is

answered May 30, 2017 by shivanip1436

2 answers

find dy/dx . if y = (x/a+)(x/b+)(x/a+)(x/b+)(...................

answered Nov 10, 2014 by pady_1

1 answer

If $ f(x) = \left\{ \begin{array}{l l} \large\frac{x^3+x^2-16x+20}{(x-2)^2} & \quad when \quad x \neq 3 \\ k & \quad when \quad x = 2 \end{array} \right. $ and $f(x)$ is continuous at $x=3$ find the value of k.

answered Apr 22, 2014 by meena.p

1 answer

If $ f(x) = \left\{ \begin{array}{l l} \large\frac{|x+2|}{\tan^{-1} (x+2)} & \quad x \neq -2 \\ 2 & \quad x=-2 \end{array} \right. $ then $f(x) $ is

answered Apr 22, 2014 by meena.p

1 answer

If the function $ f(x) = \left\{ \begin{array}{l l} 3ax+b & \quad for \quad x > 1 \\ 11 & \quad for \quad x = 1 \\ 5ax-2b & \quad for \quad x < 1 \end{array} \right. $ is continuous at $x=1$.find the values of a and b .

answered Apr 22, 2014 by meena.p

1 answer

If $ f(x+y)=f(x).f(y)$ for all x and y and if $f(5)=2$ and $f'(0) =3$ find $f'(5)$

answered Apr 22, 2014 by meena.p

1 answer

Find the values of K. So that the function $ f(x) = \left\{ \begin{array}{l l} \large\frac{k \cos x}{\pi -2x} & \quad if \quad x \neq \large\frac{\pi}{2} \\ 3 & \quad if \quad x =\large\frac{\pi}{2} \end{array} \right. $ is continuous at $x =\large\frac{\pi}{2}$ .

answered Apr 22, 2014 by meena.p

1 answer

Determine the value of K so that the function. $ f(x) = \left\{ \begin{array}{l l} Kx^2 & \quad if \quad x \leq 2 \\ 3 & \quad if \quad x > 2 \end{array} \right. $ is continuous .

answered Apr 22, 2014 by meena.p

1 answer

Evaluate $\lim \limits _{x \to \pi/6} \large\frac{(\sqrt 3 \sin x - \cos x)}{(x-\pi/6)}$

answered Apr 22, 2014 by meena.p

1 answer

Evaluate $\lim \limits_{x \to 0} \bigg(\large\frac{x^3 \cot x}{1- \cos x} \bigg)$

answered Apr 22, 2014 by meena.p

1 answer

Evaluate $\lim \limits_{x \to 0} \bigg( \large\frac{\sin 2x+ \sin 6x}{\sin 5x -\sin 3x} \bigg)$

answered Apr 22, 2014 by meena.p

1 answer

Evaluate $\lim \limits_{x \to 2} \bigg( \large\frac{e^x-e^2}{x-2} \bigg)$

answered Apr 22, 2014 by meena.p

1 answer

Define $f(0)$ so that $f(x)=(x+1)^{\cot x}$ becomes continuous at $x=0$

answered Apr 22, 2014 by meena.p

1 answer

The function $f(x)=\large\frac{|x|}{x^2+2x}\qquad$$ x\neq 0$ and $f(0)=0$ is not continuous at $x=0$ because

answered Mar 26, 2014 by balaji

1 answer

The range of the function $f(x) = \large\frac{x^2+x+2}{x^2+x+1}$, $x \in (-\infty, \infty)$ is:

answered Jan 6, 2014 by sreemathi.v

1 answer

If the function $f(x)=\left\{\begin{array}{1 1}(\cos x)^{1/x}&x\neq 0\\k&x=0\end{array}\right.$ is continuous at $x=0$ then value of $K$ is

answered Jan 6, 2014 by sreemathi.v

1 answer

The value of $f(0)$ so that $f(x)=\large\frac{(4^x-1)^3}{\sin\big(\Large\frac{x}{4}\big) \normalsize {\log}(1+\Large\frac{x^2}{3}\big)}$ is continuous everywhere is

answered Jan 6, 2014 by sreemathi.v

1 answer

let $\alpha$ and $\beta$ be the distinct roots of $ax^2+bx+c=0$ then $\lim\limits_{x\to \alpha}\large\frac{1-\cos(ax^2+bx+c)}{(x-\alpha)^2}$ is equal to

answered Jan 6, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 1}\large\frac{\sin(e^x-1)}{\log x}$ is

answered Jan 6, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to \infty}\cos(\large\frac{x}{2})$$\cos(\large\frac{x}{4})$$\cos(\large\frac{x}{8})$$........\cos(\large\frac{x}{2^n})$ is

answered Jan 6, 2014 by sreemathi.v

1 answer

If $\alpha,\beta$ are roots of $ax^2+bx+c=0$ then $\lim\limits_{x\to \alpha}\large\frac{1-\cos(ax^2+bx+c)}{(x-\alpha)^2}$ is

answered Jan 6, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to a}\bigg($ $\bigg[$ $\big(\large\frac{a^{1/2}+x^{1/2}}{a^{1/4}-x^{1/4}}\big)^{-1}$ $-$ $\frac{2(ax)^{1/4}}{x^{3/4}-a^{1/4}x^{1/2}+a^{1/2}x^{1/4}-a^{3/4}}\bigg]^{-1}$ $-$ $(\sqrt 2)^{\large\log_4 a}\bigg)^8$ is

answered Jan 6, 2014 by sreemathi.v

1 answer

The set of points where the function $f(x)=\sqrt{1-e^{-x^2}}$ is differentiable is

answered Jan 6, 2014 by sreemathi.v

1 answer

At the point $x=1$ the function $f(x)=\left\{\begin{array}{1 1}x^3-1&1 < x < \infty\\x-1&-\infty < x \leq 1\end{array}\right.$ is

answered Jan 6, 2014 by sreemathi.v

1 answer

$f(x)=\left\{\begin{array}{1 1}\large\frac{1-\cos 4x}{x^2}&x < 0\\a&x=0\\\large\frac{\sqrt x}{\sqrt{[16+\sqrt x]-4}}&x > 0\end{array}\right.$If the function be continuous at $x=0$ then $a=$

answered Jan 6, 2014 by sreemathi.v

1 answer

The function $f(x)=\left\{\begin{array}{1 1}x+a\sqrt 2\sin x&0\leq x < \large\frac{\pi}{4}\\2x\cot x+b&\large\frac{\pi}{4}\normalsize \leq x < \frac{\pi}{2}\\a\cos 2x-b\sin x&\large\frac{\pi}{2}\normalsize < x \leq \pi\end{array}\right.$ is continuous for $0 \leq x \leq \pi$ then a,b are

answered Jan 6, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{x(1+a\cos x)-b\sin x}{x^3}$$=1$ then $a,b$ are

answered Jan 6, 2014 by sreemathi.v

1 answer

If $\lim\limits_{x\to \infty}\big(1+\large\frac{a}{x}+\frac{b}{x^2}\big)^{2x}$$=e^2$ then the value of a and b are

answered Jan 6, 2014 by sreemathi.v

1 answer

Let $f(a)=g(a)=k$ and their $n^{th}$ derivatives $f^{(n)}(a),g^{(n)}(a)$ exist and are not equal for some n. Further if $\lim\limits_{x\to a}\large\frac{f(a)g(x)-f(a)-g(a)f(x)+g(a)}{g(x)-f(x)}$$=4$ then the value of K is

answered Jan 6, 2014 by sreemathi.v

1 answer

If $f(x)=\cot^{-1}\big[\large\frac{3x-x^3}{1-3x^2}\big]$ and $g(x)=\cos^{-1}\big[\large\frac{1-x^2}{1+x^2}\big]$ then $\lim\limits_{x\to a}\large\frac{f(x)-f(a)}{g(x)-g(a)}$$\quad0 < a < \large\frac{1}{2}$ is

answered Jan 6, 2014 by sreemathi.v

1 answer

If $\lim\limits_{x\to a}\large\frac{a^x-x^a}{x^x-a^a}$$=-1$ and $a > 0$ then $a$=?

answered Jan 6, 2014 by sreemathi.v

1 answer

Let $f(x+y)=f(x)f(y)$ $\forall x,y\in R$. Suppose that $f(3)=3$ and $f'(0)=11$, then $f'(3)$ is given by

answered Jan 6, 2014 by sreemathi.v

1 answer

If $\alpha$ is a repeated root of $ax^2+bx+c=0$, then $\lim\limits_{x\to \alpha}\large\frac{\sin(ax^2+bx+c)}{(x-\alpha)^2}$ =

answered Jan 6, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\bigg[\large\frac{1^x+2^x+3^x+.........n^x}{n}\bigg]^{\Large\frac{1}{x}}$ is

answered Jan 6, 2014 by sreemathi.v

1 answer

The function $f(x)=\left\{\begin{array}{1 1}\large\frac{x^2}{a}&0\leq x <1\\a&1\leq x < \sqrt 2\\\large\frac{2b^2-4b}{x^2}&\sqrt 2\leq x < \infty\end{array}\right.$ is continuous for $0\leq x < \infty$ then the most suitable values of a and b are

answered Jan 4, 2014 by sreemathi.v

1 answer

Evaluate $\lim\limits_{\large x\to a}\large\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt x}$, where $a\neq 0$

answered Jan 4, 2014 by sreemathi.v

1 answer

The value of $\lim\limits_{x\to 0}\large\frac{\sqrt{\Large\frac{1}{2}\normalsize (1-\cos 2x)}}{x}$

answered Jan 4, 2014 by sreemathi.v

1 answer

If $f(x)=\large\frac{x}{2}$$-1$, then on the interval $[0,\pi]$

answered Jan 4, 2014 by sreemathi.v

1 answer

The set of all points where the function $f(x)=\large\frac{x}{(1+|x|)}$ is differentiable is

answered Jan 4, 2014 by sreemathi.v

1 answer

Let $[x]$ denote the greatest integer less than or equal to x. If $f(x)=[x\sin\pi x]$, then $f(x)$ is

answered Jan 3, 2014 by sreemathi.v

1 answer

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

answered Jan 3, 2014 by sreemathi.v

1 answer

If $x+|y|=2y$, then $y$ as a function of x is

answered Jan 3, 2014 by sreemathi.v

1 answer

Consider the function $f(x)=|x-2|+|x-5|\;x\in R$.

answered Jan 3, 2014 by sreemathi.v

1 answer

If $f:R\to R$ is a function defined by $f(x)=[x]\cos\big(\large\frac{2x-1}{2}\big)$$\pi$ where [x] denotes the greatest integer function then f is

answered Jan 3, 2014 by sreemathi.v

1 answer

Let $f(x)=[x]\sin\bigg[\large\frac{\pi}{[x+1]}\bigg]$ where [.] denotes the greatest integer function. The domain of $f$ is and the point of discontinuity of $f$ in the domain are

answered Jan 3, 2014 by sreemathi.v

1 answer

$ABC$ is an isosceles triangle inscribed in a circle of radius r. If $AB=AC$ and $h$ is the altitude from $A$ to $BC$, then the triangle $ABC$ has perimeter $P=2(\sqrt{2hr-h^2})+\sqrt{2hr})$ and area $A$=_______also $\lim\limits_{h\to 0}\large\frac{A}{p^3}=$_______

answered Jan 3, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{x\tan 2x-2x\tan x}{(1-\cos 2x)^2}$ is

answered Jan 3, 2014 by sreemathi.v

1 answer

The function $f(x)$ = $(x^2-1)\;$ $|x^2-3x+2|$ + $\cos(|x|)$ is not differentiable at

answered Jan 3, 2014 by sreemathi.v

1 answer

The function $f(x)=[x]^2-[x^2]$ where $[x]$ is the greatest integer less than or equal to x, is discontinuous at

answered Jan 3, 2014 by sreemathi.v

1 answer

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