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How do you find the limit of #(1/x)^x# as x approaches infinity?

Answer 1

Ezra Adams

#lim_{x->infty}(1/x)^x = 0#

For #x>1-> 1/x < 1# and considering #a# such that #abs(a) < 1##lim_{x->infty}a^x = 0 #

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Answer 2

Emily Ammerman

To find the limit of (1/x)^x as x approaches infinity, we can rewrite the expression as e^(x * ln(1/x)). Using the property that ln(a/b) = ln(a) - ln(b), we can simplify it further to e^(x * (ln(1) - ln(x))). Since ln(1) is equal to 0, the expression becomes e^(-x * ln(x)). As x approaches infinity, ln(x) also approaches infinity. Therefore, the limit of (1/x)^x as x approaches infinity is 0.

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Answer 3

Isaiah Allen

The limit of ( \left(\frac{1}{x}\right)^x ) as x approaches infinity is 0.

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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.