Sequence... limit x->infinity (π^x)/(4^x)
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Sequence... limit x->infinity (π^x)/(4^x)

[From: ] [author: ] [Date: 12-03-31] [Hit: ]
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Could someone please help me with:
limit x->infinity (π^x) / (4^x)

Am I supposed to do L'Hopital's?

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limit x->infinity (π^x)/(4^x) =
limit x->infinity (π/4)^x = 0 because π/4 < 1 { definition of exponential function }

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Let:
y = lim(x-->∞) (π^x) / (4^x)

Then take the natural log of both sides:
ln(y) = lim(x-->∞) ln[(π^x) / (4^x)]
ln(y) = lim(x-->∞) ln((π/4)^x)
ln(y) = lim(x-->∞) xln(π/4)
ln(y) = -∞
y = e^(-∞)
y = 1 / e^(∞)
y = 0

Thus the limit is 0.

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