How do you evaluate #x / sqrt (4x^2 + 2x +1) # as x approaches negative infinity?

Answer 1

#-1/2#

As #x# approaches either infinity, the #2x+1# becomes increasing an monotonically smaller compared to the #4x^2#. So ignore it to find the limit: #x/sqrt(4x^2)=x/(2x)=1/2#. A more rigorous procedure would involve the delta-epsilon construction. However, in cases of simple rational fractions, or something close to them like this, all you need do, usually, is ignore all powers except the highest in the numerator and denominator.
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Answer 2

As x approaches negative infinity, the expression x / sqrt(4x^2 + 2x + 1) can be evaluated by considering the dominant term in the denominator. In this case, the dominant term is 4x^2. As x approaches negative infinity, 4x^2 also approaches positive infinity. Therefore, the expression x / sqrt(4x^2 + 2x + 1) approaches 0.

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Answer from HIX Tutor

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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