How do you find the end behavior of #f (x) = 3x^4  5x + 1#?
See the explanation below.
First, graph the function. graph{3x^45x+1 [10, 10, 5, 5]}
You can also write this using limits:
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To find the end behavior of a polynomial function like (f(x) = 3x^4  5x + 1), you look at the term with the highest degree, which is the (3x^4) term in this case.

When the leading term has an even exponent (like (x^4)), the end behavior as (x) approaches positive or negative infinity is the same sign as the leading coefficient. So, as (x) approaches positive infinity, (f(x)) also approaches positive infinity, and as (x) approaches negative infinity, (f(x)) also approaches positive infinity.

When the leading term has an odd exponent (like (x^3)), the end behavior as (x) approaches positive infinity is the same sign as the leading coefficient, and as (x) approaches negative infinity, (f(x)) approaches negative infinity.
Therefore, for the function (f(x) = 3x^4  5x + 1), as (x) approaches positive infinity, (f(x)) approaches positive infinity, and as (x) approaches negative infinity, (f(x)) approaches positive infinity.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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