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What is the limit of #(1+4/x)^x# as x approaches infinity?

Answer 1

Emma Aldridge

#lim_{x->oo}(1 + 4/x)^x = e^4#

Notice that

#(1 + 4/x)^x = e^(x ln(1 + 4/x))#

and if the limit exists,

#lim_{x -> oo} ( e^(x ln(1 + 4/x)) ) = e^{lim_{x -> oo}(x ln(1+4/x))}#

as the exponential function is continuous everywhere.

To evaluate the limit at the exponent, we first write it as

#x ln(1 + 4/x) = frac{ln(1 + 4/x)}{1/x}#

Since the form is indeterminate #0/0#, use the L'hospital rule.

#lim_{x->oo}(ln(1+4/x)/(1/x)) = lim_{x->oo}(frac{frac{d}{dx}(ln(1+4/x))}{frac{d}{dx}(1/x)})#

#= lim_{x->oo}(frac{-4/x^2}{(1+4/x)}/(-1/x^2))#

#= lim_{x->oo}(4/(1+4/x))#

#= frac{4}{1+0}#

#= 4#

Therefore, the limit is #e^4#.

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Answer 2

Emily Ammerman

If you are familiar with the sometimes definition of #e#, as #lim_(urarroo)(1+1/u)^u = e#, then we don't need l"Hospital.

#lim_(xrarroo)(1+4/x)^x = lim_(xrarroo)(1+1/(x/4))^(4(x/4))#

# = lim_(xrarroo)((1+1/(x/4))^(x/4))^4#

Now, with #u = x/4#, we have

# = lim_(urarroo)((1+1/u)^u)^4#

# = (lim_(urarroo)(1+1/u)^u)^4 = e^4#

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Answer 3

Emma Aldridge

The limit of (1+4/x)^x as x approaches infinity is e^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.