If $f(x) = \frac{x}{x - 1}$,then $\frac{f(a)}{f(a + 1)} = $

If $f(x + y) = f(x) + f(y)$ for all $x, y \in R$ and $f(1) = 1$,then find the value of $\lim_{x \to 0} \frac{2^{f(\tan x)} - 2^{f(\sin x)}}{f(\tan x) - f(\sin x)}$.

Let $R$ denote the set of all real numbers. Let $a_i, b_i \in R$ for $i \in \{1, 2, 3\}$. Define the functions $f: R \rightarrow R$,$g: R \rightarrow R$,and $h: R \rightarrow R$ by $f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4$ and $g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4$. Let $h(x) = f(x+1) - g(x+2)$. If $f(x) \neq g(x)$ for every $x \in R$,then the coefficient of $x^3$ in $h(x)$ is:

Let $f: R \rightarrow R$ be such that $f$ is injective and $f(x)f(y) = f(x+y)$ for all $x, y \in R$. If $f(x), f(y),$ and $f(z)$ are in $GP$, then $x, y,$ and $z$ are in:

If $f$ is a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in N$ such that $f(1)=3$ and $\sum\limits_{x = 1}^n {f(x) = 120}$,find the value of $n$.

The values of $b$ and $c$ for which the identity $f(x + 1) - f(x) = 8x + 3$ is satisfied,where $f(x) = bx^2 + cx + d$,are: