Limits and Cont...

Suppose the function $$f(x)-f(2x)$$ has the derivative $$5$$ at $$x=1$$ and derivative $$7$$ at $$x=2$$. The derivative of the function $$f(x)-f(4x)$$ at $$x=1$$, has the value equal to?

$$\lim _{ x\rightarrow 0 }{ \log _{ \left( \tan ^{ 2 }{ x }  \right)  }{ \left( \tan ^{ 2 }{ 2x }  \right) = }  }$$

$$\displaystyle \lim_{n \rightarrow \infty} {^{n}C_{c}}\left(\dfrac {m}{n}\right)^{x}\left(1-\dfrac {m}{n}\right)^{n-x}$$ equal to

If [.] denotes, GIF , then $$\underset{x \rightarrow 0}{lt} \left( \left[\dfrac{2018 sin^{-1} x}{x}\right] + \left[\dfrac{2020x}{tan^{-1} x}\right]\right)$$ = 

The value of $$\lim\limits_{x\to 0}\dfrac{1-\cos x}{x^2}$$

$$\displaystyle\mathop {\lim }\limits_{\ x \to 0} \cos \frac{x}{2}\cos \frac{x}{{{2^2}}}\cos \frac{x}{{{2^3}}}......\cos \frac{x}{{{2^n}}}$$ is equal to 

$$\underset{h \rightarrow 0}{lim} \dfrac{\sqrt{x + h} -\sqrt{x}}{h}$$ is equal to 

Evaluate:

$$\lim\limits_{x\to 0}\dfrac{1-\cos x }{x^2}=$$

$$\underset{x \to 0}{\lim}\dfrac{\sin [ \cos x]}{1+[\cos x]}$$ is