Question

please don't post unnecessary answers

$\bold{lim_{x \to{0} \dfrac{x-\sqrt{x^{2}+x}}{x} }   }$  ​

Answer 1

$\bold{lim_{x \to{0}} \dfrac{x-\sqrt{x^{2}+x}}{x}   }$ 

If we put x=0 here,it is indeterminate form =0/0

so,

$\sf{lim_{x \to{0}} \dfrac{x-\sqrt{x^{2}+x}}{x} × \dfrac{x+\sqrt{x^{2}+x}}{x+\sqrt{x^{2}+x}}}$

$\tt{lim_{x \to{0}} \dfrac{x^{2}-(x^{2}+x)}{ x(x+\sqrt{x^{2}+x} )} }$

$\sf{lim_{x \to{0}} \dfrac{-x}{ x(x+\sqrt{x^{2}+x} )} }$

$\tt{lim_{x \to{0}} \dfrac{-1}{ x+\sqrt{x^{2}+x} } }$

Now put x=0 ,we get that,

• $\footnotesize{\bold{\blue{\dfrac{-1}{0}  }} }$ 

which is not finite

Hence,

The limit is not defined