If $y=\sqrt{x}+\frac{1}{\sqrt{x}},$ then $\frac{dy}{dx}$ at $x=1$ is

Select the correct answer from the given alternatives.

$\underset{x\to 2}{\lim }\left(\frac{x^4-16}{x^2-5x+6}\right)=$

Select the correct answer from the given alternatives.

$\underset{x\to 5}{\lim }\left(\frac{\sqrt{x+4}-3}{\sqrt{3x-11}-2}\right)=$

The value of $\underset{x\to 3}{\lim }\left(\frac{1}{x^2-11x+24}+\frac{1}{x^2-x-6}\right)=$

Select the correct answer from the given alternatives.

$\underset{x\to \frac{\pi }{3}}{\lim }\left(\frac{{\tan }^{2}x-3}{{\sec }^{3}x-8}\right)=$

Mark the correct alternative in the following:

$\underset{n\to \infty }{\lim }\frac{1^2+2^2+3^2+\dots +n^2}{n^3}$ is equal to.