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

Answer 1

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

Ok this requires a few wee tricks. We want to find

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

Because the exponential and natural log functions are inverse to each other they cancel out so we can rewrite this as

#lim_(xrarroo) exp(ln((x+1)/(x))^x)#

Using rules of logs we can bring the exponent down:

#lim_(xrarroo) exp(xln((x+1)/(x)))#

Now notice that the bit that actually changes is the exponent of the exponential function, so that's what we focus on:

#exp(lim_(xrarroo) (xln((x+1)/(x))))#
If at all possible we want to use L'hopital's rule on the limit. For this we need it to be in indeterminate form, eg #0/0 or (oo)/(oo) etc..#

Rewrite as:

#exp(lim_(xrarroo) (ln((x+1)/(x))/(1/x)))#

Check the numerator and denominator separately:

#lim_(xrarroo) 1/x = 0#
#lim_(xrarroo) ln((x+1)/x) = lim_(xrarroo) ln((1+1/x)/1) = ln(1) = 0#
so we have #0/0# and are in the correct form for L'hopital's.
#exp(lim_(xrarroo) ((d/(dx)(ln((x+1)/(x))))/(d/(dx)(1/x))))#

Computing the derivative of the numerator using chain and quotient rules:

#d/(dx)(ln((x+1)/x)) = x/(x+1)*(-1)/(x^2) = - 1/(x(x+1))#

For the denominator, easiest just to rewrite and use power rule

#d/(dx)( x^(-1)) = -1/x^2#

We now have:

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

So we end up with

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

The limit of ((x+1)/x)^x as x approaches infinity is e, where e is the mathematical constant approximately equal to 2.71828.

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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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