Limits — NDA Mathematics PYQs

What is ${\lim }_{x\to 1}\frac{x^{(n^2-1)}-1}{x^{(n+1)}-1}$ equal to, where $n>1$ is a natural number ?

What is ${\lim }_{x\to 0}\frac{10^{\sin x}-1}{\tan x}$ equal to ?

Consider the following for the two (02) items that follow :
Let $f(x)=\{\begin{array}{ll}{x}^3,& x^2<1\\ x^2,& x^2\ge 1\end{array}$

What is $\underset{x\to 0}{\lim }{f}^{\prime }(x)$ equal to?

Consider the following for the two (02) items that follow :
Let $f(x)=\sin [x]$, where $[\cdot ]$ is the greatest integer function and $g(x)=|x|$.

What is $\underset{x\to 0}{\lim }\{f(x)\cdot g(x)\}$ equal to?

Consider the following for the two (02) items that follow :
Let $f(x)=\sin [x]$, where $[\cdot ]$ is the greatest integer function and $g(x)=|x|$.

What is $\underset{x\to 0}{\lim }\frac{f(x)}{g(x)}$ equal to?

Consider the following for the two (02) items that follow :
Let the function $f(x)=x^2-1$.

What is $\underset{x\to 1}{\lim }\{f\circ f(x)\}$ equal to?

Consider the following for the two (02) items that follow :
Let the function $f(x)=x^2+9$.

What is $\underset{x\to 0}{\lim }\frac{\sqrt{f(x)}-3}{\sqrt{f(x)+7}-4}$ equal to?

Directions : For the following two (02) items :
Let
$$f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2}& ,x<0\\ 9& ,x=0\\ \frac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}& ,x>0\end{array}$$

What is ${\lim }_{x\to 0^{-}}f(x)$ equal to?

Directions : For the following two (02) items :
Let
$$f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2}& ,x<0\\ 9& ,x=0\\ \frac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}& ,x>0\end{array}$$

What is ${\lim }_{x\to 0^{+}}f(x)$ equal to?

Directions : For the following three (03) items :
Consider the function $f(x)=x|x|$.

What is ${\lim }_{x\to -1}f(x)$ equal to?