Question

Find the limit as x approaches infinity of (x+(x)^1/2)^1/2-(x)^1/2

Answer 1

\[(\color{red}{x}+x^{1/2})^\color{red}{{1/2}}-\color{red}{x^{1/2}}\]

if this is what you mean you can re-arrange it to:

\[= x^{1/2} [(1+\dfrac{1}{x^{1/2}})^{1/2}-1]\]

and if you want expand out the bit on the left .... Binomial series expansion


\[= \color{blue}{x^{1/2}} \left[ \left(\color{red}{1}+\frac{1}{2} \dfrac{1}{\color{blue}{x^{1/2}}} + \frac{1}{2} (-\frac{1}{2}) \frac{1}{2!}\dfrac{1}{x} + \dots \right)-\color{red}{1} \right]\]

with \(x \to \infty\)

Answer 2

So the limit should be equal to 1/2

Answer 3

i'd say so!