# How do you evaluate the limit #(x^4+3^x)/(x^5+1)# as x approaches #oo#?

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To evaluate the limit as x approaches infinity of (x^4+3^x)/(x^5+1), we can use the concept of dominant terms.

As x approaches infinity, the term with the highest power in the numerator and denominator will dominate the expression. In this case, the dominant terms are x^4 in the numerator and x^5 in the denominator.

Dividing both the numerator and denominator by x^5, we get (x^4/x^5 + 3^x/x^5)/(x^5/x^5 + 1/x^5).

Simplifying further, we have (1/x + (3^x)/(x^5))/(1 + 1/x^5).

As x approaches infinity, 1/x approaches 0, and (3^x)/(x^5) also approaches 0. Therefore, the expression simplifies to (0 + 0)/(1 + 0), which is equal to 0.

Hence, the limit of (x^4+3^x)/(x^5+1) as x approaches infinity is 0.

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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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