How do you find the end behavior of #f(x) = –x^4 – 4#?

Answer 1

#-infty#

We know that any polynomial will tend to positive or negative infinity.

We can determine which one by the sign of the coefficient in front of the leading term. Here, the leading term is #-x^4# and since that coefficient is negative, the function will limit to negative infinity.

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Answer 2

James Allen

To find the end behavior of the function (f(x) = -x^4 - 4), examine the leading term of the polynomial, which is (-x^4).

As (x) approaches infinity ((x \to \infty)), the term (-x^4) dominates, leading (f(x)) to approach negative infinity ((f(x) \to -\infty)). This indicates that the right end of the graph descends without bound.
As (x) approaches negative infinity ((x \to -\infty)), the term (-x^4) also dominates and, due to the negative coefficient, leads (f(x)) again to approach negative infinity ((f(x) \to -\infty)). This means the left end of the graph also descends without bound.

Therefore, the end behavior of (f(x) = -x^4 - 4) is:

As (x \to \infty), (f(x) \to -\infty)
As (x \to -\infty), (f(x) \to -\infty)

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Answer from HIX Tutor

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.