decide whether the following statement is true or false. if the degree of the numerator of a rational…

decide whether the following statement is true or false. if the degree of the numerator of a rational function equals the degree of the denominator, then the ratio of the leading coefficients gives rise to the horizontal asymptote. choose the correct answer below. false true

decide whether the following statement is true or false. if the degree of the numerator of a rational function equals the degree of the denominator, then the ratio of the leading coefficients gives rise to the horizontal asymptote. choose the correct answer below. false true

Answer

Explanation:

Step1: Recall rational - function rule

For a rational function $f(x)=\frac{a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_0}{b_nx^n + b_{n - 1}x^{n - 1}+\cdots+b_0}$, where $n$ is the degree of the numerator and denominator ($n\geq0$).

Step2: Analyze horizontal asymptote

As $x\to\pm\infty$, we divide both the numerator and denominator by $x^n$. Then $f(x)=\frac{a_n+\frac{a_{n - 1}}{x}+\cdots+\frac{a_0}{x^n}}{b_n+\frac{b_{n - 1}}{x}+\cdots+\frac{b_0}{x^n}}$. As $x\to\pm\infty$, the terms with $\frac{1}{x},\frac{1}{x^2},\cdots,\frac{1}{x^n}$ approach 0. So, $\lim_{x\to\pm\infty}f(x)=\frac{a_n}{b_n}$, which is the horizontal asymptote.

Answer:

True