Recent questions tagged ch13

$\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

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

asked 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

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

The function $f(x)=[x]\cos\big[\large\frac{2x-1}{2}\big]$$\pi$ where [.] denotes the greatest integer function, is discontinuous at

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

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

If $f(x)$ is continuous and differentiable function and $f(1/n)=0\forall n \geq 1$ and $n\in 1$ then

asked Dec 31, 2013 by sreemathi.v

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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