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Recent questions tagged limits and derivatives

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

asked Apr 22, 2014 by meena.p

1 answer

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

asked Apr 22, 2014 by meena.p

1 answer

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

asked Apr 22, 2014 by meena.p

1 answer

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

asked Apr 22, 2014 by meena.p

1 answer

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

asked Apr 22, 2014 by meena.p

1 answer

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

asked 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

asked Mar 26, 2014 by balaji

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

asked 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

asked 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

asked Jan 6, 2014 by sreemathi.v

1 answer

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

asked 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

asked 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

asked 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

asked 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

asked 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

asked 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 $a$=?

asked 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

asked Jan 6, 2014 by sreemathi.v

1 answer

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

asked 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

asked Jan 6, 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 $a\neq 0$

asked Jan 4, 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 $\lim\limits_{h\to 0}\large\frac{A}{p^3}=$_______

asked 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

asked Jan 3, 2014 by sreemathi.v

1 answer

If $\lim\limits_{x\to \infty}\bigg[\large\frac{x^2+x+1}{x+1}$$-ax-b\bigg]=4$ then

asked Jan 3, 2014 by sreemathi.v

1 answer

The value of $\lim\limits_{x\to 0}\big((\sin x)^{1/x}+(1+x)^{\large\sin x}\big)$ where $x > 0$ is

asked Jan 3, 2014 by sreemathi.v

1 answer

$\lim\limits_{n\to \infty}\left\{\large\frac{1}{1-n^2} + \frac{1}{1-n^2} +......+ \large\frac{n}{1-n^2}\right\}$ is equal to

asked Jan 3, 2014 by sreemathi.v

1 answer

The function $f(x)=\large\frac{ln(1+ax)-ln(1-bx)}{x}$ is not defined at $x=0$. The value which should be assigned to $f$ at $x=0$ so that it is continuous at $x=0$ is

asked Jan 3, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 1}(\log_22x)^{\large\log_x5}$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{(\cos x)^{1/2}-(\cos x)^{1/3}}{\sin^2x}$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

If $a,b,c,d$ are positive then $\lim\limits_{x\to \infty}\big(1+\large\frac{1}{a+bx}\big)^{c+dx}=$

asked Jan 2, 2014 by sreemathi.v

1 answer

If $f(x)$ is the integral function of the function $\large\frac{2\sin x-\sin 2x}{x^3}\qquad$$ x\neq 0$ then $\lim\limits_{x\to 0} f'(x)$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\big(\large\frac{x^2+5x+3}{x^2+x+2}\big)^x$ =

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{2^x-1}{\sqrt{1+x}-1}$=

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 1}\big[\sec\big(\large\frac{\pi x}{2}\big)$$\log x\bigg]$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{(1+x)^{1/x}-e}{x}$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0^+}x^m(\log x)^n,(m,n\in N)$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

The value of $\lim\limits_{n\to \infty}x\bigg[\tan^{-1}\large\frac{x+1}{x+2}$$-\cot^{-1}\large\frac{x+2}{x}\bigg]$ is

asked Jan 2, 2014 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\large\frac{\sin x}{x}$

asked Dec 31, 2013 by sreemathi.v

1 answer

Find $\lim\limits_{x\to 0}\{\tan(\large\frac{\pi}{4}$$+x)\}^{1/x}$

asked Dec 31, 2013 by sreemathi.v

1 answer

Use the formula $\lim\limits_{x\to 0}\large\frac{a^x-1}{x}=$$ln\; a$ to find $\lim\limits_{x\to 0}\large\frac{2^x-1}{(1+x)^{1/2}-1}$

asked Dec 31, 2013 by sreemathi.v

1 answer

$f(x)$ is the integral of $\large\frac{2\sin x-\sin 2x}{x^3}$$x\neq 0$ find $\lim\limits_{x\to 0}f'(x)$

asked Dec 31, 2013 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)+f(a)}{g(x)-f(x)}$$=4$, then the value of k is

asked Dec 31, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to \infty}\big[\large\frac{x^2+5x+3}{x^2+x+3}\big]^x$

asked Dec 31, 2013 by sreemathi.v

1 answer

The left hand derivative of $f(x)=[x]\sin(\pi x)$ at $x=k$, where $k$ is an integer is

asked Dec 31, 2013 by sreemathi.v

1 answer

If $f(x)=\left\{\begin{array}{1 1}\large\frac{\sin [x]}{[x]}&[x]\neq 0\\0&[x]=0\end{array}\right.$, where $[x]$ denotes the greatest integer less than or equal to x, then $\lim\limits_{x\to 0}f(x)$ equals

asked Dec 31, 2013 by sreemathi.v

1 answer

If $f(a)=2$, $f'(a)=1$, $g(a)=-1$, $g'(a)=2$, then the value of $\lim\limits_{x\to a}\large\frac{g(x)f(a)-g(a)f(x)}{x-a}$ is

asked Dec 31, 2013 by sreemathi.v

1 answer

$\lim\limits_{h\to 0}\large\frac{ln(1+2h)-2ln(1+h)}{h^2}=$

asked Dec 30, 2013 by sreemathi.v

1 answer

$\lim\limits_{x\to 0}\bigg[\large\frac{1+5x^2}{1+3x^2}\bigg]^{\Large\frac{1}{x^2}}$

asked Dec 30, 2013 by sreemathi.v

1 answer

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

asked Dec 30, 2013 by sreemathi.v

1 answer

If $f(x)=\large\frac{1}{\sqrt{18-x^2}}$ then $\lim\limits_{x\to 3}\large\frac{f(x)-f(3)}{x-3}$ is

asked Dec 30, 2013 by sreemathi.v

1 answer

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