Determine the end behavior of each function below.

For $f(x)=\frac{-18 x^2+13 x-13}{5 x^2-10}$:

As $x \rightarrow \infty, y \rightarrow$ □

As $x \rightarrow-\infty, y \rightarrow$ □

For $f(x)=\frac{-3 x^5-11}{4 x^3+15 x+7}$:

As $x \rightarrow \infty, y \rightarrow$ □

As $x \rightarrow-\infty, y \rightarrow$ □

For $f(x)=\frac{16 x^2-3}{9 x^6+19 x+19}$:

As $x \rightarrow \infty, y \rightarrow$ □

As $x \rightarrow-\infty, y \rightarrow$ □

For $f(x)=\frac{10 x^3-19}{7 x^2+7}$:

As $x \rightarrow \infty, y \rightarrow$ □

As $x \rightarrow-\infty, y \rightarrow$ □

Question

06-11-2026
Mathematics

Answered

Determine the end behavior of each function below.

For $f(x)=\frac{-18 x^2+13 x-13}{5 x^2-10}$:

As $x \rightarrow \infty, y \rightarrow$ □

As $x \rightarrow-\infty, y \rightarrow$ □

For $f(x)=\frac{-3 x^5-11}{4 x^3+15 x+7}$:

As $x \rightarrow \infty, y \rightarrow$ □

As $x \rightarrow-\infty, y \rightarrow$ □

For $f(x)=\frac{16 x^2-3}{9 x^6+19 x+19}$:

As $x \rightarrow \infty, y \rightarrow$ □

As $x \rightarrow-\infty, y \rightarrow$ □

For $f(x)=\frac{10 x^3-19}{7 x^2+7}$:

As $x \rightarrow \infty, y \rightarrow$ □

As $x \rightarrow-\infty, y \rightarrow$ □

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T23:34:12-05:00

Function 1 ($n=m$): Both limits are $\boxed{-3.6}$.
Function 2 ($n>m$): Both limits are $\boxed{-\infty}$.
Function 3 ($nm$): The limit as $x \to \infty$ is $\boxed{\infty}$ and as $x \to -\infty$ is $\boxed{-\infty}$.

Explanation

Analytical Framework for End Behavior

To find the end behavior of a rational function $f(x) = \frac{P(x)}{Q(x)}$, we compare the degree of the numerator $n$ and the degree of the denominator $m$. As $x \to \pm\infty$, the function behaves like the ratio of its leading terms.

If $n = m$, the function has a horizontal asymptote at the ratio of the leading coefficients.
If $n < m$, the horizontal asymptote is $y = 0$.
If $n > m$, the function approaches $\pm\infty$ depending on the behavior of the leading terms' ratio.

Analyzing the First Function

For $f(x) = \frac{-18x^2 + 13x - 13}{5x^2 - 10}$, the degrees are $n=2$ and $m=2$. The leading terms are $-18x^2$ and $5x^2$. As $x \to \pm\infty$, the ratio approaches $\frac{-18}{5} = -3.6$. Therefore, the limits as $x \to \infty$ and $x \to -\infty$ are both $-3.6$.

Analyzing the Second Function

For $f(x) = \frac{-3x^5 - 11}{4x^3 + 15x + 7}$, the degrees are $n=5$ and $m=3$. The leading term ratio is $\frac{-3x^5}{4x^3} = -\frac{3}{4}x^2$. As $x \to \infty$, $-\frac{3}{4}x^2 \to -\infty$. As $x \to -\infty$, $-\frac{3}{4}x^2$ still approaches $-\infty$ because $x^2$ is positive. Both limits are $-\infty$.

Analyzing the Third Function

For $f(x) = \frac{16x^2 - 3}{9x^6 + 19x + 19}$, the degrees are $n=2$ and $m=6$. Since the denominator's degree is greater than the numerator's ($n < m$), the ratio $\frac{16x^2}{9x^6} = \frac{16}{9x^4}$ approaches $0$ as $|x| \to \infty$. Both limits are $0$.

Analyzing the Fourth Function

For $f(x) = \frac{10x^3 - 19}{7x^2 + 7}$, the degrees are $n=3$ and $m=2$. The leading term ratio is $\frac{10x^3}{7x^2} = \frac{10}{7}x$. As $x \to \infty$, the function approaches $\infty$. As $x \to -\infty$, the function approaches $-\infty$.

Examples

Understanding the end behavior of rational functions is vital in fields like electrical engineering and control systems. For example, when analyzing the frequency response of a filter circuit, the gain of the circuit often follows a rational function. Knowing the end behavior allows engineers to predict if the system will become unstable or if the signal will be suppressed (tending toward zero) or amplified indefinitely as frequency increases, ensuring safe and predictable circuit performance.