# How do you find the limit #lim (3^(x+5)-2^(2x+1))/(3^(x+1)-2^(2x+4))# as #x->oo#?

Rewriting:

Concentrating on the dominating terms of numerator and denominator:

Dividing:

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To find the limit as x approaches infinity, we can analyze the exponents of the terms involved. In the numerator, the exponent of 3 is (x+5), while the exponent of 2 is (2x+1). In the denominator, the exponent of 3 is (x+1), and the exponent of 2 is (2x+4).

As x approaches infinity, the terms with smaller exponents become negligible compared to those with larger exponents. Therefore, we can ignore the terms involving 2 in both the numerator and denominator.

This simplifies the expression to (3^(x+5))/(3^(x+1)).

Using the properties of exponents, we can rewrite this as 3^(x+5-x-1), which simplifies to 3^4.

Thus, the limit of the given expression as x approaches infinity is 81.

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