Question

limx tends to zero 1+2+3+....+x /x*2​

Answer 1

Step-by-step explanation:

$\lim _{x - > 0} \frac{1 + 2 + 3 + .... + x}{ {x}^{2} }$

$\lim _{x - > 0} \frac{ \frac{x(x + 1)}{2} }{ {x}^{2} }$

$\lim _{x - > 0} \frac{x(x + 1)}{2 {x}^{2} }$

$\lim _{x - > 0} \frac{x + 1}{2x}$

This in 0/0 form, so, using l'hospital's rule,

$\lim _{x - > 0} \frac{ \frac{d}{dx}(x + 1) }{ \frac{d}{dx} (2x)}$

$\lim _{x - > 0} \frac{1}{2}$

$= \frac{1}{2}$