# How do you find the limit of #(3/x)^(1/x)# as x approaches infinity?

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To find the limit of (3/x)^(1/x) as x approaches infinity, we can rewrite the expression using the properties of logarithms. Taking the natural logarithm of both sides, we get ln((3/x)^(1/x)). Using the logarithmic property ln(a^b) = b * ln(a), we can rewrite this as (1/x) * ln(3/x). Simplifying further, we have ln(3/x)/x.

Now, we can apply the limit properties. As x approaches infinity, 3/x approaches 0, and ln(3/x) approaches ln(0), which is undefined. However, we can rewrite ln(3/x) as ln(3) - ln(x) using the logarithmic property ln(a/b) = ln(a) - ln(b).

So, the expression becomes (ln(3) - ln(x))/x. As x approaches infinity, ln(x) also approaches infinity. Therefore, we have (ln(3) - infinity)/infinity, which is an indeterminate form.

To evaluate this indeterminate form, we can apply L'Hôpital's rule. Taking the derivative of the numerator and denominator separately, we get (-1/x)/(1). Simplifying, we have -1/x.

Now, as x approaches infinity, -1/x approaches 0. Therefore, the limit of (3/x)^(1/x) as x approaches infinity is 1.

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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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