HomerSimpson
New member
- Jun 12, 2008
- 14
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Need: limit as x -> infinity of [sqrt(x^2 - x) - sqrt(x^2 - x - 2)]
Clearly the limit is zero since the -2 will become insignificant as x becomes very large. But i'd like to answer the question without this explanation (i know its valid but still seems dodgy to me).
I changed sqrt(x^2 - x) - sqrt(x^2 - x - 2) to
{sqrt(x^2 - x)} * {1 - sqrt[1 - 2/(x^ - x)]}
So then the term 2/(x^ - x) approaches 0 as x approaches infinity. And so {1 - sqrt[1 - 2/(x^ - x)]} would become 1-1=0. But then as x approaches infinity {sqrt(x^2 - x)} approaches infinity. So then we have infinity*0, which is undefined. So then this method doesn't seem to give me a definite answer. Can someone point out any mistakes and show how i can use this method to show clearly and definitely that the limit is zero or provide another method which will allow me to do so.
Clearly the limit is zero since the -2 will become insignificant as x becomes very large. But i'd like to answer the question without this explanation (i know its valid but still seems dodgy to me).
I changed sqrt(x^2 - x) - sqrt(x^2 - x - 2) to
{sqrt(x^2 - x)} * {1 - sqrt[1 - 2/(x^ - x)]}
So then the term 2/(x^ - x) approaches 0 as x approaches infinity. And so {1 - sqrt[1 - 2/(x^ - x)]} would become 1-1=0. But then as x approaches infinity {sqrt(x^2 - x)} approaches infinity. So then we have infinity*0, which is undefined. So then this method doesn't seem to give me a definite answer. Can someone point out any mistakes and show how i can use this method to show clearly and definitely that the limit is zero or provide another method which will allow me to do so.