Questions: The graph of f(x)=(-3 x^4 -2 x+5)/(x^5 +x^2 -2) will behave like which function for large values of x?

Solution Steps

To determine the behavior of the function \( f(x) = \frac{-3x^4 - 2x + 5}{x^5 + x^2 - 2} \) for large values of \(|x|\), we need to analyze the degrees of the polynomial in the numerator and the denominator. The degree of the numerator is 4, and the degree of the denominator is 5. For large \(|x|\), the function will behave like the ratio of the leading terms of the numerator and the denominator, which is \(\frac{-3x^4}{x^5} = -\frac{3}{x}\).

We start with the function

\[ f(x) = \frac{-3x^4 - 2x + 5}{x^5 + x^2 - 2}. \]

To understand its behavior as \( |x| \) becomes large, we need to identify the leading terms in both the numerator and the denominator.

The leading term in the numerator is \( -3x^4 \) and in the denominator, it is \( x^5 \). Thus, for large values of \( |x| \), we can approximate the function as:

\[ f(x) \approx \frac{-3x^4}{x^5} = -\frac{3}{x}. \]

To find the behavior as \( x \) approaches infinity, we compute the limit:

\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} -\frac{3}{x} = 0. \]

Final Answer