Consider a function $f(x)$ which has exactly two roots at $x=a$. If $\mathop {\lim }\limits_{x \to a}\left(\frac{\lambda f^{\prime}(x)}{f(x)}-\frac{1}{x-a}\right)=m(\neq 0)$, then the value of $\lambda$ ix

If $a=\mathop {\lim }\limits_{n \to \infty } \cos ^{2 n} x,(x=n \pi)$ and $b=\mathop {\lim }\limits_{n \to \infty } \cos ^{2 n} x,(x \neq n \pi)$, then numerical value of the area of the triangle whose vertices are (a, b), (-2, 1) and (2, 1) is

For a real number $y$, consider $(y)$ denotes the greatest integer less than or equal to $y$. If $f(x)=\frac{\tan (\pi[x-\pi])}{1+[x]^2}$, then

If $f(x)$ is differentiable for all $x \in \mathbb{R}$ and satisfies the relation

$x=\mathop {\lim }\limits_{n \to \infty }\frac{\left[1^2(f(x))^x\right]+\left[2^2(f(x))^x\right]+\ldots+\left[n^2(f(x))^x\right]}{n^3}$ where [.] denotes the greatest integer function, then $f^{\prime}(x)=$