Question

solve the following limit.​

Answer 1

Step-by-step explanation:

We have,

$\lim_{x \rarr0} \frac{ \sqrt{ {x}^{2} + x + 4 } - 2}{x}$

$= \lim_{x \rarr0} \frac{( \sqrt{ {x}^{2} + x + 4 } - 2)( \sqrt{ {x}^{2} + x + 4 } + 2)}{x( \sqrt{ {x}^{2} + x + 4 } + 2)}$

$= \lim_{x \rarr0} \frac{ {x}^{2} + x + 4 - 4 }{x( \sqrt{ {x}^{2} + x + 4 } + 2)}$

$= \lim_{x \rarr0} \frac{x(x + 1)}{x( \sqrt{ {x}^{2} + x + 4 } + 2) }$

$= \lim_{x \rarr0} \frac{(x + 1)}{( \sqrt{ {x}^{2} + x + 4 } + 2)}$

$= \frac{(0 + 1)}{( \sqrt{0 + 0 + 4} + 2)}$

$= \frac{1}{4}$