Limit Calculation

Nov 2005
127
4
Glasgow, Scotland
Hi guys, I need a little help with this limit here:

lim x->infinity of: sqrt(x^2 + x) - sqrt(x^2 - 2x)

cheers !
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Hi guys, I need a little help with this limit here:

lim x->infinity of: sqrt(x^2 + x) - sqrt(x^2 - 2x)

cheers !
"Rationalize the numerator"- that is, multiply both numerator and denominator by \(\displaystyle \sqrt{x^2+ x}+ \sqrt{x^2- 2x}\) to get \(\displaystyle \frac{(\sqrt{x^2+ x}- \sqrt{x^2- 2x})(\sqrt{x^2+ x}+ \sqrt{x^2- 2x})}{\sqrt{x^2+ x}+ \sqrt{x^2- 2x}}\)\(\displaystyle = \frac{x^2+ x- x^2+ 2x}{\sqrt{x^2+x}+ \sqrt{x^2- 2x}}= \frac{3x}{\sqrt{x^2+ x}+ \sqrt{x^2- 2x}}\)
 
Nov 2005
127
4
Glasgow, Scotland
done it, cheers !
(Clapping)
 

simplependulum

MHF Hall of Honor
Jan 2009
715
427
Or we can solve it by mean value theorem ,

Let \(\displaystyle f(x) = \sqrt{t} \)
we have

\(\displaystyle \frac{ \sqrt{x^2 + x } - \sqrt{x^2 - 2x}}{x^2 + x - x^2 + 2x} = \frac{1}{2\sqrt{c}} \) where \(\displaystyle x^2 - 2x < c < x^2 + x\)

Take reciporcal and square root ,

\(\displaystyle \frac{1}{\sqrt{x^2 - 2x}} > \frac{1}{\sqrt{c}} > \frac{1}{\sqrt{x^2 +x }}\)

\(\displaystyle \frac{1}{\sqrt{x^2 - 2x}} > 2 \frac{ \sqrt{x^2 + x } - \sqrt{x^2 - 2x}}{3x}> \frac{1}{\sqrt{x^2 +x }} \)


\(\displaystyle \frac{3}{2} \frac{x}{\sqrt{x^2 - 2x}} > \sqrt{x^2 + x } - \sqrt{x^2 - 2x} > \frac{3}{2} \frac{x}{\sqrt{x^2 +x }} \)

\(\displaystyle \lim_{x\to\infty} \frac{3}{2} \frac{x}{\sqrt{x^2 - 2x}}> \lim_{x\to\infty} \sqrt{x^2 + x } - \sqrt{x^2 - 2x} > \lim_{x\to\infty} \frac{3}{2} \frac{x}{\sqrt{x^2 +x }} \)

\(\displaystyle \frac{3}{2} > \lim_{x\to\infty} \sqrt{x^2 + x } - \sqrt{x^2 - 2x} >\frac{3}{2} \)


\(\displaystyle L = \frac{3}{2} \)
 
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