MHB Solve Limit with Square Root: \[\lim_{x\rightarrow -\infty }\sqrt{x^{2}+3}+x\]

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To solve the limit \[\lim_{x\rightarrow -\infty }\sqrt{x^{2}+3}+x\], the approach involves multiplying by the conjugate \(\frac{\sqrt{x^2+3}-x}{\sqrt{x^2+3}-x}\). This technique simplifies the expression, allowing for the cancellation of terms. The limit ultimately evaluates to 0, as the dominant terms balance each other out. The discussion highlights the importance of using algebraic manipulation to clarify the limit's behavior. Understanding this method provides a clear technical demonstration of the limit's value.
Yankel
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Hello

I am trying to solve this limit here:

\[\lim_{x\rightarrow -\infty }\sqrt{x^{2}+3}+x\]

I understand that it should be 0 since the power and square root cancel each other, while the power turned the minus into plus, and then when I add infinity I get 0. This is logic, I wish to know how to show it technically, if possible.

Thank you.
 
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Multiply by $\frac{\sqrt{x^2+3}-x}{\sqrt{x^2+3}-x}$.
 
Thank you, I didn't see it. :o
 

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