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How do you find the limit of # (3 x^4 + 4) / ((x^2 - 7)(4 x^2 - 1)) # as x approaches infinity?

Answer 1

Charles Arocha

#3/4#

# (3 x^4 + 4) / ((x^2 - 7)(4 x^2 - 1))=x^4/x^4(3+4/x^4)/((1-7/x^2)(4-1/x^2))# so

#lim_(x->oo) (3 x^4 + 4) / ((x^2 - 7)(4 x^2 - 1)) = lim_(x->oo)(3+4/x^4)/((1-7/x^2)(4-1/x^2)) = 3/4#

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Answer 2

Isaiah Allen

To find the limit of the given expression as x approaches infinity, we can analyze the highest power terms in the numerator and denominator. In this case, the highest power terms are x^4 in the numerator and x^2 in the denominator.

Dividing both the numerator and denominator by x^4, we get:

(3 x^4 + 4) / ((x^2 - 7)(4 x^2 - 1)) = (3 + 4/x^4) / ((1 - 7/x^2)(4 - 1/x^2))

As x approaches infinity, the terms 4/x^4, 7/x^2, and 1/x^2 all tend towards zero. Therefore, the expression simplifies to:

(3 + 0) / (1 - 0)(4 - 0) = 3 / (1)(4) = 3/4

Hence, the limit of the given expression as x approaches infinity is 3/4.

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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.