Limit problem

[manorange]

I would like to know why is this way to solve this limit incorrect. I've plotted the function, and I get that it doesn't go like this, but I don't know what rule has been broken.

 

[Steven G]

The answer to your question is that [math]\infty*0\neq0[/math]
It is possible that the small difference in those parenthesis you have when multiplied by [math]\infty[/math] can equal something other than 0.

Sure 0 times any well defined number is 0 BUT 0 times an undefined number (like infinity) is not necessary 0 !!!

 

[pka]

You have the incorrect simplification.
\(\sqrt{x^2+x}-x=\dfrac{x^2}{\sqrt{x^2+x}+x}=\dfrac{1}{\sqrt{1+\dfrac{1}{x}}+1}\)
Now \(\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} + x} - x} \right)=?\) SEE HERE