How do you find the limit of #(2u+1)^4/(3u^2+1)^2# as #u->oo#?

Answer 1

# lim_(u rarr oo)(2u+1)^4/(3u^2+1)^2 = 16/9#

# lim_(u rarr oo)(2u+1)^4/(3u^2+1)^2 = lim_(u rarr oo) ((2u)^4 + 4(2u)^3 + 6(2u)^2+4(2u)+1)/((3u^2)^2+2(3u^2)+1) #
# :. lim_(u rarr oo)(2u+1)^4/(3u^2+1)^2 = lim_(u rarr oo) (16u^4 + 32u^3 + 24u^2+8u+1)/(9u^4+6u^2+1) #
# :. lim_(u rarr oo)(2u+1)^4/(3u^2+1)^2 = lim_(u rarr oo) (16u^4 + 32u^3 + 24u^2+8u+1)/(9u^4+6u^2+1) * (1/u^4)/(1/u^4)#
# :. lim_(u rarr oo)(2u+1)^4/(3u^2+1)^2 = lim_(u rarr oo) (16 + 32/u + 24/u^2+8/u^3+1/u^4)/(9+6/u^2+1/u^4)# # :. lim_(u rarr oo)(2u+1)^4/(3u^2+1)^2 = 16/9#

This is supported by the following graph:

graph{ [-3.25, 16.75, -4.48, 5.52]} / (2x+1)^4/(3x^2+1)^2

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Answer 2

#lim_(urarroo)(2u+1)^4/(3u^2+1)^2=16/9#

The trick with these is to factor out the greatest degree of #u# that we can from the numerator and denominator:
#lim_(urarroo)(2u+1)^4/(3u^2+1)^2=lim_(urarroo)[u(2+1/u)]^4/[u^2(3+1/u^2)]^2#
#color(white)(lim_(urarroo)(2u+1)^4/(3u^2+1)^2)=lim_(urarroo)(u^4(2+1/u)^4)/((u^2)^2(3+1/u^2)^2)#
#color(white)(lim_(urarroo)(2u+1)^4/(3u^2+1)^2)=lim_(urarroo)(2+1/u)^4/(3+1/u^2)^2#
As #urarroo#, or as #u# becomes increasingly large, we see that #1/u# and #1/u^2# get smaller and smaller denominators up to the point where #1/u,1/u^2rarr0#.
#color(white)(lim_(urarroo)(2u+1)^4/(3u^2+1)^2)=(2)^4/(3)^2#
#color(white)(lim_(urarroo)(2u+1)^4/(3u^2+1)^2)=16/9#
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Answer 3

To find the limit of (2u+1)^4/(3u^2+1)^2 as u approaches infinity, we can use the concept of limits.

First, we divide both the numerator and denominator by u^4, which allows us to focus on the highest power terms.

This simplifies the expression to (2/u + 1/u^4)^4 / (3 + 1/u^2)^2.

As u approaches infinity, the terms with 1/u and 1/u^4 become negligible compared to the other terms.

Thus, the limit of the expression is (2/u)^4 / 3^2, which simplifies to 16/9.

Therefore, the limit of (2u+1)^4/(3u^2+1)^2 as u approaches infinity is 16/9.

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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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