# How do you find the limit of # ln ( 3x + 5e^x )/ ln ( 7x + 3e^{2x})# as x approaches infinity?

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To find the limit of ln(3x + 5e^x) / ln(7x + 3e^(2x)) as x approaches infinity, we can use the properties of logarithms and the concept of limits.

First, let's simplify the expression by applying the logarithmic identity ln(a) - ln(b) = ln(a/b):

ln(3x + 5e^x) / ln(7x + 3e^(2x)) = ln[(3x + 5e^x) / (7x + 3e^(2x))]

Next, we can use the fact that the natural logarithm of a product is equal to the sum of the natural logarithms of the individual factors:

ln[(3x + 5e^x) / (7x + 3e^(2x))] = ln(3x + 5e^x) - ln(7x + 3e^(2x))

Now, as x approaches infinity, the term 5e^x and 3e^(2x) will dominate the expressions 3x and 7x respectively. This is because the exponential functions grow much faster than the linear functions as x becomes larger.

Therefore, we can simplify the expression further by neglecting the terms 3x and 7x:

ln(3x + 5e^x) - ln(7x + 3e^(2x)) ≈ ln(5e^x) - ln(3e^(2x))

Using the logarithmic identity ln(a^b) = b * ln(a), we can rewrite the expression:

ln(5e^x) - ln(3e^(2x)) = ln(5) + ln(e^x) - ln(3) - ln(e^(2x))

Since ln(e^x) = x and ln(e^(2x)) = 2x, we have:

ln(5) + ln(e^x) - ln(3) - ln(e^(2x)) = ln(5) + x - ln(3) - 2x

Combining like terms, we get:

ln(5) + x - ln(3) - 2x = -ln(3) - x + ln(5)

As x approaches infinity, the terms -x and -ln(3) become negligible compared to ln(5). Therefore, the limit of ln(3x + 5e^x) / ln(7x + 3e^(2x)) as x approaches infinity is ln(5).

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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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