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

Binomial expansion on the radical

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Rewrite the expression using the conjugate.

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To find the limit of sqrt(x^2 + 1) - 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 sqrt(x^2 + 1) + x. This will help eliminate the square root in the numerator. After simplifying, we get (sqrt(x^2 + 1) - x) * (sqrt(x^2 + 1) + x) / (sqrt(x^2 + 1) + x). Simplifying further, we have (x^2 + 1 - x^2) / (sqrt(x^2 + 1) + x). This simplifies to 1 / (sqrt(x^2 + 1) + x). As x approaches infinity, the denominator also approaches infinity, so the limit of the expression is 1 / infinity, which is equal to 0. Therefore, the limit of sqrt(x^2 + 1) - x as x approaches infinity is 0.

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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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