Limits and Deri...

Let $$f(x)$$ be a polynomial function of second degree.If $$f(1)=f(-1)$$ and  $$a,b,c$$ are in A.P $$ f'(a)$$,$$f'(b)$$,$$f'(c)$$ are in 

If $$\phi (x) =\displaystyle \lim_{n \rightarrow \infty} \frac{x^{2n} f(x) + g(x)}{1 + x^{2n}}$$, then

Let $$f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}}, 1 \le x \le 26$$ be a real valued function, then $$f'(x)$$ for $$1 < x < 26$$ is

Let $$f$$ be a differentiable function satisfying $$f(x) + f(y) + f(z) + f(x)f(y)f(z) = 14$$ for all $$x,\space y,\space z \in R$$
Then,

Given  $$f(x)=-\displaystyle \frac{x^3}{3}+x^2\sin 1.5a-x\sin a.\sin 2a-5 \arcsin (a^2-8a+17)$$ then :

Suppose, $$A=\displaystyle \frac {dy}{dx}$$ of $$x^2+y^2=4$$ at $$(\sqrt 2, \sqrt 2), B=\displaystyle \frac {dy}{dx}$$ of $$sin y+sin x=sin x\cdot sin y$$ at $$(\pi, \pi)$$ and $$C=\displaystyle \frac {dy}{dx}$$ of $$2e^{xy}+e^xe^y-e^x=e^{xy+1}$$ at $$(1, 1)$$, then $$(A-B-C)$$ has the value equal to .....

Suppose $$A=\displaystyle \frac{dy}{dx}$$ when $$x^2+y^2=4$$ at $$(\sqrt{2},\sqrt{2})$$,$$ B=\displaystyle \frac{dy}{dx}$$ when $$\sin y+ \sin x=\sin x-\sin y$$ at $$(\pi,\pi)$$ and $$C=\displaystyle  \frac{dy}{dx}$$ when $$2e^{xy}+e^x e^y-e^x-e^y=e^{xy+1}$$ at $$(1,1)$$, then $$(A+B+C)$$ has the value equal to 

If $$\displaystyle\lim_{x\rightarrow a}{(f(x)+g(x))}=2$$ and $$\displaystyle\lim_{x\rightarrow a}{(f(x)-g(x))}=1$$, 

Let f and g be differentiable function such that $${f}'\left ( x \right )=2g\left ( x \right )$$ and $${g}'\left ( x \right )=-f\left ( x \right )$$, and let $$T\left ( x \right )=\left ( f\left ( x \right ) \right )^{2}-\left ( g\left ( x \right ) \right )^{2}$$. Then $${T}'\left ( x \right )$$ is equal to

The value of $$\displaystyle \underset { n\rightarrow \infty  }{ lim } \left( \frac { 1 }{ n+1 } +\frac { 1 }{ n+2 } +...+\frac { 1 }{ 6n }  \right) $$ is