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How do you find the limit of #((x^2)-2x+3)/(6-(3x^4))# as x approaches #oo#?

Answer 1

Ezra Adams

#((x^2)-2x+3)/(6-(3x^4)) |_{x to oo}#

#((1/x^2)-2/x^3+3/x^4)/(6/x^4-(3)) |_{x to oo}#

#0/(-3) = 0#

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Answer 2

Christopher Agresta

To find the limit of ((x^2)-2x+3)/(6-(3x^4)) as x approaches infinity, we can analyze the highest power terms in the numerator and denominator. In this case, the highest power term in the numerator is x^2, and the highest power term in the denominator is -3x^4.

As x approaches infinity, the term -3x^4 dominates the expression. Therefore, we can simplify the expression by dividing both the numerator and denominator by x^4. This yields:

((x^2)/x^4 - (2x)/x^4 + 3/x^4) / (6/x^4 - (3x^4)/x^4)

Simplifying further, we get:

(1/x^2 - 2/x^3 + 3/x^4) / (6/x^4 - 3)

As x approaches infinity, all terms with x in the denominator tend to zero. Thus, the expression simplifies to:

(0 - 0 + 0) / (0 - 3)

Finally, we have:

0 / (-3) = 0

Therefore, the limit of ((x^2)-2x+3)/(6-(3x^4)) as x approaches infinity is 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.