# How do you find the limit of #((e^x)-x)^(2/x)# as x approaches infinity?

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To find the limit of ((e^x)-x)^(2/x) as x approaches infinity, we can use the properties of exponential and logarithmic functions. By taking the natural logarithm of both sides, we can simplify the expression. Applying the limit laws, we can then evaluate the limit.

Taking the natural logarithm of ((e^x)-x)^(2/x), we get: ln(((e^x)-x)^(2/x))

Using the properties of logarithms, we can bring the exponent down: (2/x) * ln((e^x)-x)

Now, we can evaluate the limit as x approaches infinity.

By applying the limit laws, we can simplify the expression further: lim(x→∞) (2/x) * ln((e^x)-x) = 2 * lim(x→∞) (1/x) * ln((e^x)-x)

Since the limit of (1/x) as x approaches infinity is 0, we have: 2 * 0 * ln((e^x)-x) = 0

Therefore, the limit of ((e^x)-x)^(2/x) as x approaches infinity is 0.

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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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