Limits — NDA Mathematics PYQs

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?

Consider the following for the two (02) items that follow :
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 $\underset{x\to 0^{-}}{\lim }f(x)$ equal to?

Consider the following for the two (02) items that follow :
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 $\underset{x\to 0^{+}}{\lim }f(x)$ equal to?

Consider the following for the three (03) items that follow :
Consider the function $f(x)=x|x|$.

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