Limits and Cont...

Let $$f : R\rightarrow R$$ be any function, Define
$$g:R\rightarrow R$$ by $$g(x)=|f(x)|\forall x$$, then

$$\displaystyle \lim_{x\rightarrow \infty }\frac{2x+7\sin x}{4x+3\cos x}=$$

The function $$y=3\sqrt{x}-|x-1|$$ is continuous at

 lf $$\mathrm{f}(\mathrm{x})=\left\{\begin{matrix}1+x &x\leq 1 \\  3-ax^{2}& x>1\end{matrix}\right.$$ is continuous at $${x}=1$$ then $${a}=({a}>0)$$

The function $$\displaystyle \mathrm{f}({x})=\frac{1+\sin x-\cos x}{1-\sin x-\cos x}$$ is not defined at $${x}=0$$. The value of $$\mathrm{f}(\mathrm{0})$$ so that $$\mathrm{f}({x})$$ is continuous at $${x}=0$$ is

$$\displaystyle Lt_{x\rightarrow 0^+}(sinx)^{\tan x}=$$

lf the function $$\displaystyle \mathrm{f}({x})=\frac{e^{x^{2}}-\cos {x}}{x^{2}}$$ for $$x \neq 0$$ is continuous at $${x}=0$$ then $$\mathrm{f}(\mathrm{0})=$$

$$\displaystyle \lim_{\mathrm{x}\rightarrow \pi }(1- 4 \tan \mathrm{x} )^{\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{x}}=$$

$$\displaystyle \lim_{n\rightarrow \infty }(\pi n)^{2/n}=$$

$$\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} \displaystyle\frac { { x }^{ 5 }-32 }{ x-2 } , & x\neq 2 \end{matrix} \\ \begin{matrix} k, & x=2 \end{matrix} \end{cases}$$ is continuous at $$x=2$$, then the value of $$k$$ is