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

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

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

Using rules of logs we can bring the exponent down:

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

Rewrite as:

Check the numerator and denominator separately:

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

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

We now have:

So we end up with

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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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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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