How do you evaluate the limit #sqrt(x^2+1)-2x# as x approaches #oo#?
See below.
Change the way the expression is written.
Therefore,
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To evaluate the limit of sqrt(x^2+1)-2x as x approaches infinity, we can simplify the expression. By multiplying the numerator and denominator by the conjugate, sqrt(x^2+1)+2x, we can eliminate the square root in the numerator. This simplifies the expression to (sqrt(x^2+1)-2x)(sqrt(x^2+1)+2x) / (sqrt(x^2+1)+2x).
Expanding the numerator gives (x^2+1) - (4x^2) = -3x^2 + 1.
As x approaches infinity, the -3x^2 term dominates, and the limit becomes -3x^2 / (sqrt(x^2+1)+2x).
Since the degree of the numerator is greater than the degree of the denominator, the limit as x approaches infinity is negative infinity.
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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.
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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