# How do you find the limit of #[(x^2+x)^(1/2)-x]# as x approaches infinity?

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So, finally, we get:

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To find the limit of [(x^2+x)^(1/2)-x] as x approaches infinity, we can simplify the expression by multiplying both the numerator and denominator by the conjugate of the expression, which is [(x^2+x)^(1/2)+x]. This will help eliminate the square root in the numerator. Simplifying further, we get [(x^2+x) - x^2] / [(x^2+x)^(1/2)+x]. Canceling out the x^2 terms, we are left with x / [(x^2+x)^(1/2)+x]. As x approaches infinity, the x term dominates the expression, and the denominator becomes insignificant compared to x. Therefore, the limit of [(x^2+x)^(1/2)-x] as x approaches infinity is 1.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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