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 June 4th, 2012, 03:24 PM #1 Senior Member   Joined: Jan 2012 Posts: 159 Thanks: 0 As The Limit Goes To Infinity... taking the conjugate of the numerator seems long, am I missing something?
 June 4th, 2012, 06:23 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: As The Limit Goes To Infinity... We are given: $L=\lim_{x\to\infty}\frac{x-\sqrt{x^2+5x+2}}{x-\sqrt{x^2+\frac{x}{2}+1}}$ I would write: $\frac{x-\sqrt{x^2+5x+2}}{x-\sqrt{x^2+\frac{x}{2}+1}}\cdot\frac{\frac{1}{x}}{\ frac{1}{x}}=\frac{1-\sqrt{1+\frac{5}{x}+\frac{2}{x^2}}}{1-\sqrt{1+\frac{1}{2x}+\frac{1}{x^2}}}$ Now we have: $L=\lim_{x\to\infty}\frac{1-\sqrt{1+\frac{5}{x}+\frac{2}{x^2}}}{1-\sqrt{1+\frac{1}{2x}+\frac{1}{x^2}}}$ Which is the the indeterminate form 0/0, thus L'Hôpital's rule gives (after simplification): $L=\lim_{x\to\infty}\frac{2(5x+4)\sqrt{1+\frac{1}{2 x}+\frac{1}{x^2}}}{(x+4)\sqrt{1+\frac{5}{x}+\frac{ 2}{x^2}}}=10$
 June 4th, 2012, 06:56 PM #3 Senior Member   Joined: Jan 2012 Posts: 159 Thanks: 0 Re: As The Limit Goes To Infinity... Is there a way to solve without L'Hôpital's rule? I am just studying past tests but our teach said we cannot use L'Hôpital's rule but maybe previous years could... Dunno
 June 4th, 2012, 07:57 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: As The Limit Goes To Infinity... Okay, barring the use of our good friend L'Hôpital, we could complete the square under the radicals to get: $L=\lim_{x\to\infty}\frac{x-\sqrt{$$x+\frac{5}{2}$$^2-\frac{17}{4}}}{x-\sqrt{$$x+\frac{1}{4}$$^2+\frac{3}{4}}}$ Now, we may observe that: $\lim_{x\to\infty}\frac{\sqrt{(x+a)^2+k}}{\sqrt{(x+ a)^2}}=1$ where $k\in\mathbb R$ hence: $\lim_{x\to\infty}\sqrt{(x+a)^2+k}=\lim_{x\to\infty }\sqrt{(x+a)^2}$ thus, we have: $L=\lim_{x\to\infty}\frac{x-\sqrt{$$x+\frac{5}{2}$$^2}}{x-\sqrt{$$x+\frac{1}{4}$$^2}}=\frac{-\frac{5}{2}}{-\frac{1}{4}}=10$
 June 4th, 2012, 08:11 PM #5 Senior Member   Joined: Jan 2012 Posts: 159 Thanks: 0 Re: As The Limit Goes To Infinity... nice mark I tried completing the square but made a simple mistake! At least I was headed in the right direction.

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