Question

1.7 - 1.10 quiz ap precalc - no calculator. not eligible for reassessment
name:
date: 12/3/25
period: lsh

1. determine the end - behavior of the following.
2. f(x)=(4x^(3)+1)/(3x^(2)+2x+3)
3. h(x)=(x^(2)-9)/(2x+1)

end behavior:
end behavior:

1. evaluate the following limit.

lim_(x rarr -oo)(4x^(3)-x^(2))/(2x^(2)+9)

1. use the graph of f(x) to answer the following.

end behavior:
is there a horizontal asymptote? if so, write the equation of the horizontal asymptote.

1. find the zeroes of the following rational functions.

f(x)=(x^(2)-5x - 14)/(x - 3)

Explanation:

Step1: Determine end - behavior of \(f(x)=\frac{4x^3 + 1}{3x^2+2x + 3}\)

As \(x\to\pm\infty\), the end - behavior of a rational function is determined by the leading terms of the numerator and denominator. The degree of the numerator \(n = 3\) and the degree of the denominator \(m=2\). Since \(n>m\), as \(x\to\infty\), \(f(x)\to\infty\) and as \(x\to-\infty\), \(f(x)\to-\infty\).

Step2: Determine end - behavior of \(h(x)=\frac{24}{x^2 - 9}\)

Factor the denominator \(x^2 - 9=(x + 3)(x - 3)\). As \(x\to\pm\infty\), the function \(h(x)\to0\) because the degree of the denominator (\(m = 2\)) is greater than the degree of the numerator (\(n=0\)).

Step3: Evaluate \(\lim_{x\to-\infty}\frac{4x^3-7x^2}{2x^2 + 9}\)

Divide both the numerator and denominator by \(x^2\) (the highest - power of \(x\) in the denominator). We get \(\lim_{x\to-\infty}\frac{4x-7}{2+\frac{9}{x^2}}\). As \(x\to-\infty\), \(\frac{9}{x^2}\to0\), so \(\lim_{x\to-\infty}\frac{4x-7}{2+\frac{9}{x^2}}=-\infty\).

Step4: Find zeros of \(f(x)=\frac{x^2-5x - 14}{x - 3}\)

Set the numerator equal to zero: \(x^2-5x - 14=(x - 7)(x+2)=0\). So \(x = 7\) and \(x=-2\) are the zeros of the function (since the denominator is non - zero at these points).

Step5: Find horizontal asymptote of \(f(x)\)

For a rational function \(\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n>m\), there is no horizontal asymptote. If \(n = m\), the horizontal asymptote is \(y=\frac{a_n}{b_m}\), and if \(n

Answer:

End - behavior of \(f(x)=\frac{4x^3 + 1}{3x^2+2x + 3}\): as \(x\to\infty\), \(f(x)\to\infty\); as \(x\to-\infty\), \(f(x)\to-\infty\)
End - behavior of \(h(x)=\frac{24}{x^2 - 9}\): as \(x\to\pm\infty\), \(h(x)\to0\)
\(\lim_{x\to-\infty}\frac{4x^3-7x^2}{2x^2 + 9}=-\infty\)
Zeros of \(f(x)=\frac{x^2-5x - 14}{x - 3}\): \(x = 7\) and \(x=-2\)
Horizontal asymptote analysis depends on the rational function's numerator and denominator degrees as described above.