# How do you determine the limit of #5^x/(3^x+2^x)# as x approaches infinity?

Use l'Hopital's Rule and some algebra.

In the original form, l'Hopital's rule doesn't help, but note that:

L'Hopital tells us to consider the ratio of the derivatives:

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To determine the limit of 5^x/(3^x+2^x) as x approaches infinity, we can use the concept of limits and properties of exponential functions.

First, let's rewrite the expression as (5/3)^x / (1 + (2/3)^x).

As x approaches infinity, (5/3)^x will tend to infinity since the base (5/3) is greater than 1.

Similarly, (2/3)^x will tend to 0 as x approaches infinity since the base (2/3) is between 0 and 1.

Therefore, the limit of (5/3)^x / (1 + (2/3)^x) as x approaches infinity is infinity.

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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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