Limits and Cont...

$$\displaystyle \lim_{x\rightarrow \frac{\pi }{4}}\displaystyle \frac{\cos x-\sin x}{(\frac{\pi}{4}-x)(\cos x+\sin x)}$$=

$$\displaystyle \lim_{x\rightarrow \dfrac{\pi }{2}}\displaystyle \frac{1-\sin x}{(\pi-2x)^{2}}$$=

$$\displaystyle \lim_{x\rightarrow \infty }\frac{x+\sin x}{x+ \cos x}=$$

$$ \displaystyle f(x)=\left\{ \begin{matrix} \dfrac { 3 }{ { x }^{ 2 } } \sin { 2{ x }^{ 2 } } ,\; x<0 \\ \dfrac { { x }^{ 2 }+2x+C }{ 1-3{ x }^{ 2 } } ,\; x\ge 0,\; x\neq \frac { 1 }{ \sqrt { 3 }  }  \\ 0,\; x=\frac { 1 }{ \sqrt { 3 }  }  \end{matrix} \right. $$
$$\displaystyle \lim_{x\rightarrow 0^{+}}f(x)=$$

Let $$f(x)=\begin{cases}x^{2}-1,0 < x < 2\\2x+3,2 \leq x < 3\end{cases},$$ 

The value of $$\sqrt{e}$$e upto four decimal places is?

If $$f(x) = \displaystyle \frac {x^2+6x}{\sin x}$$ , then $$\displaystyle \lim_{x\rightarrow 0^{-}}f(x)=$$

$$\displaystyle \lim_{x\rightarrow 0}\displaystyle \frac{\log(1+ax)-\log(1+bx)}{x}$$=

Let $$f$$ be a continuous function on [1,3]. lf $$f$$ takes only rational values for all $$x$$ and $$f(2)=10$$ then $$f(1.5)$$ is equal to

$$\displaystyle \lim _{ x\rightarrow { 0 }^{ - } }{ \dfrac { 3\sin(2{ x }^{ 2 }) }{ { x }^{ 2 } }  } =A$$