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What is the limit of #(1 + 2/x)^x# as x approaches infinity?

Answer 1

Christopher Agresta

#lim_(xrarroo)(1+2/x)^x = e^2#

Use #lim_(urarroo)(1+1/u)^u = e#

#lim_(xrarroo)(1+2/x)^x = lim_(xrarroo)(1+1/(x/2))^x#

# = lim_(xrarroo)((1+1/(x/2))^(x/2))^2#

# = (lim_(xrarroo)(1+1/(x/2))^(x/2))^2#

Now we have the form above with #u = x/2#, so we evaluate the limit.

# = e^2#

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Answer 2

Cameron Alexander

Jim, awesome ..

What this limit really represents is essentially the horizontal asymptote y = #e^2#, reflecting the function's long term graphical behavior.

Here are a couple of TI screenshots showing the graph and the decimal expansion for #e^2#.

But in fact you are looking at the curve endowed with concavity, not a straight line. The curve is asymptotically approaching the value of #y=e^2#

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Answer 3

Samantha Adkison

The limit of (1 + 2/x)^x as x approaches infinity is e^2.

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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.