Limits — Class 12 Mathematics PYQs

Consider the following for the next two (02) items that follow:

Let $f(x)=|x|+1$ and $g(x)=[x]-1$, where $[.]$ is the greatest integer function. Let $h(x)=\frac{f(x)}{g(x)}$

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

Consider the following for the next two (02) items that follow :

Let $f(x)=\frac{a^{x-1}+b^{x-1}}{2}$ and $g(x)=x-1$.

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

Consider the following for the next two (02) items that follow :

Let $f(x)=\frac{a^{x-1}+b^{x-1}}{2}$ and $g(x)=x-1$.

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

Consider the following for the next two (02) items that follow :

Let $f(x)=|x|$ and $g(x)=[x]-1$, where $[.]$ is the greatest integer function.
Let $h(x)=\frac{f(g(x))}{g(f(x))}$.

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

Consider the following for the next two (02) items that follow :

Let $f(x)=|x|$ and $g(x)=[x]-1$, where $[.]$ is the greatest integer function.
Let $h(x)=\frac{f(g(x))}{g(f(x))}$.

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

Consider the following for the next two (02) items that follow :

Let $f(x)=\{\begin{array}{ll}ax(x+1)+b,& x<1\\ x-1,& 1\le x\le 2\end{array}$

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

Consider the following for the next three (03) items that follow :

Let $f(x)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$

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

Consider the following for the next three (03) items that follow :

Let $f(x)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$

What is ${\lim }_{x\to 0}\frac{f(x)}{x^2}$ equal to?

What is ${\lim }_{x\to 5}\frac{5-x}{|x-5|}$ equal to?

What is ${\lim }_{x\to 1}\frac{x^9-1}{x^3-1}$ equal to?