Evaluate the limit $\underset{x\to \pi }{\lim }sgn\left[\tan x\right]$ (where $\left[.\right]$ represents the greatest integer function)

Evaluate the limit $\underset{x\to 1}{\lim }{\sin }^{-1}\left(\ln \left(x\right)\right)$

If $f\left(x\right)=\left\{\begin{array}{cc}x+1& , x<1\\ 2x-3& , x\ge 1\end{array}\right.,$ evaluate $\underset{x\to 1}{\lim }f\left(\mathit{x}\right)$

Let $f\left(x\right)=\left\{\begin{array}{ll}x+\lambda & , x<1\\ 2x-3& , x\ge 1\end{array}\right.,$ if $\underset{x\to 1}{\lim }f\left(x\right)$ exist, then find value of $\mathrm{\lambda }.$

Which of the followings are indeterminate forms. Also state the type.

$\left(\mathrm{i}\right)$ $\underset{x\to 0^{+}}{\lim }\frac{\left[\mathrm{x}\right]}{\mathrm{x}}$, where $\left[.\right]$ denotes the greatest integer function

$\left(\mathrm{ii}\right)$ $\underset{x\to -\infty }{\lim }\sqrt{x^2+1}-x$

$\left(\mathrm{iii}\right)$ $\underset{x\to {\left(\frac{\mathrm{\pi }}{2}\right)}^{-}}{\lim }{\left(\tan x\right)}^{\tan 2x}$

(iv) $\underset{\mathrm{x}\to 1^{+}}{\mathrm{\ell im}},{\left(\left\{\mathrm{x}\right\}\right)}^{\frac{1}{\mathrm{\ell nx}}}$ where $\left\{.\right\}$ denotes the fractional part function

Evaluate each of the following limits, if exists

 $\underset{x\to -1}{\lim }\frac{x^3-3x+1}{x-1}$

Evaluate each of the following limit, if exists

$\underset{x\to 1}{\lim }\frac{4x^3-x^2+2x-5}{x^6+5x^3-2x-4}$

Evaluate each of the following limit, if exists

 $\underset{x\to a}{\lim }\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}, a\ne 0$