Rewriting Limit Problem with Conjugate - Explained | Get a Better Understanding

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The discussion centers on evaluating the limit of the expression \lim_{x \to \infty} (\sqrt{x^2 + 1} - 1). Participants clarify that substituting 0 for x is incorrect since the limit approaches infinity, not zero. The recommended approach involves rewriting the expression using its conjugate to simplify the limit calculation. This method is essential for accurately determining the limit as x approaches infinity.

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x[itex]\stackrel{Lim}{\rightarrow}[/itex]infinity ([itex]\sqrt{x^{2}+1}[/itex])[itex]-1[/itex]

Couldn't I just inset 0 for x and end up getting [itex]\sqrt{1}[/itex]?

The book tells me to rewrite this problem by its conjugate.

My question is why? Thanks
 
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CrossFit415 said:
x[itex]\stackrel{Lim}{\rightarrow}[/itex]infinity ([itex]\sqrt{x^{2}+1}[/itex][itex]-1[/itex]

Couldn't I just inset 0 for x and end up getting [itex]\sqrt{1}[/itex]?

The book tells me to rewrite this problem by the conjugate.

My question is why? Thanks
As you have written it,
[tex]\lim_{x \to \infty}\sqrt{x^2 + 1} - 1 = \infty[/tex]
You seem to be a little confused in this problem. Why would you put in zero for x when the limit is as x goes to infinity?
 

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