Hello,
Answer D
\(\sqrt{x-5} - \sqrt{x} =5\)
==>\((x-5) -2*\sqrt{(x-5)x}+x =25\)
==>\(2x-30=\sqrt{x(x-5)}\)
==>\(x-15=\sqrt{x(x-5)}\)
==>\(x^2-30x+225=x(x-5)\)
==>\(225=-5x+30x\)
==>\(25x=225\)
==>\(x=9\)
But:
\(\sqrt{9-5} - \sqrt{9}=\sqrt{4} - \sqrt{9}=2-3=-1\)≠5
Explanation:
\(\lim_{x \to \infty} \sqrt{x-5}- \sqrt{x}\)
\(= \lim_{x \to \infty} \frac{(\sqrt{x-5}- \sqrt{x})*(\sqrt{x-5}+ \sqrt{x})}
{\sqrt{x-5}+ \sqrt{x}}\)
\(= \lim_{x \to \infty} \frac{(-5)}{\sqrt{x-5}+ \sqrt{x}}\)
\(= \lim_{x \to \infty} \frac{-5}{\infty}=0\)
