What is the limit of #(1-7/x)^x# as x approaches infinity?

Answer 1

It is #1/e^7#.

We are tempted to substitute and say that #7/x# goes to zero so it is #1# to the infinity that is #1#. But this is wrong because it is enough that the quantity in the parenthesis is slightly different from #1# and the big exponent does the rest.
So it is better to transform it in something that remove the #x# from the exponent. The best trick is to use the logarithm. So I study the limit of the logarithm and in the end I will do the exponential of the solution.
#lim_(x->oo)ln(1-7/x)^x# #=lim_(x->oo)xln(1-7/x)# #=lim_(x->oo)ln(1-7/x)/x^-1#
Now the limit is in the form of #0/0# and I can apply the rule of Hôpital evaluating the derivative:
#lim_(x->oo)ln(1-7/x)/x^-1# #=lim_(x->oo)(d/dxln(1-7/x))/(d/dxx^-1)#
#=lim_(x->oo) (7/((x-7)x))/(-x^-2)#
#=lim_(x->oo) -(7x^2)/((x-7)x)# #=lim_(x->oo) -(7x)/((x-7))#
This form is still #0/0# so I re-apply Hôpital's rule
#lim_(x->oo) -(7x)/((x-7))#
#=lim_(x->oo) -(d/dx7x)/(d/dx(x-7))#
#=lim_(x->oo) -7/1=-7#

This is the limit of the logarithm, so the final result is obtained doing the exponential of this:

#lim_(x->oo)(1-7/x)^x=e^-7=1/e^7#.
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Answer 2

The limit of (1-7/x)^x as x approaches infinity is 1/e^7.

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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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