Factorial — Question 34

The domain of the function $\left[ 2tan\left( \pi x \right) \right]^{{log_{\left[ 2tan\left( \pi x \right) \right]}}\left( g\left( x \right)\right)}$, where $g(x)=\left( \frac{x^{2}+2x-3}{4x^{2}-4x-3} \right)$ contain the interval $[n+\frac{1}{4},n+\frac{1}{2})$, where $n$ is an integer. Then $n$ can be equal to (Note: [.] denotes greatest integer function)

If unit digit of $a!+b!+c!$ is 9 then find the value of ${(a!)(b!)(c!)}$

Find the number of negative integral values of $m$ for which the expression $x^{2}+2\left( m-1 \right)x+m+5$ is positive $\forall$ $x \gt 1$.

A function $f:R\to R$ is defined as $f\left( x \right)=3x+5$. Find $f^{-1}\left( x \right)$.

Let $f:R\to \left( 0,\infty \right)$ be such that $f\left( x \right)+\frac{e^{x+x^{2}}}{f\left( x \right)}\le e^{x}+e^{x^{2}}, \forall$ $x>0$. Then, $\displaystyle\lim_{x \to 1} f\left( x \right)$ is