# How do you find the limit of #(2x^3+3x^2cosx)/(x+2)^3# as #x->oo#?

The limit has indeterminate initial form

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To find the limit of (2x^3+3x^2cosx)/(x+2)^3 as x approaches infinity, we can use the concept of dominant terms.

First, we observe that as x approaches infinity, the cosine term oscillates between -1 and 1, and its effect becomes negligible compared to the polynomial terms.

Next, we focus on the dominant terms, which are the highest power terms in the numerator and denominator. In this case, it is 2x^3 in the numerator and (x+2)^3 in the denominator.

By dividing both the numerator and denominator by x^3, we can simplify the expression to (2 + 3cosx/x)/(1 + 2/x)^3.

As x approaches infinity, the cosine term becomes insignificant, and we are left with (2 + 0)/(1 + 0)^3, which simplifies to 2/1^3, or simply 2.

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

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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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