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Find the horizontal asymptote.

[tex]\ \textless \ br/\ \textgreater \ \begin{array}{l}\ \textless \ br/\ \textgreater \ y=\frac{4x + 32}{x - 8} \\\ \textless \ br/\ \textgreater \ y=[?]\ \textless \ br/\ \textgreater \ \end{array}\ \textless \ br/\ \textgreater \ [/tex]

Sagot :

GinnyAnsweridn
To find the horizontal asymptote of the rational function [tex]\( y = \frac{4x + 32}{x - 8} \)[/tex], follow these steps:

1. Identify the degrees of the numerator and the denominator:

The highest degree term in the numerator is [tex]\(4x\)[/tex], which is degree 1. The highest degree term in the denominator is [tex]\(x\)[/tex], which is also degree 1.

2. Compare the degrees:

Both the numerator and the denominator have the same degree, which is 1.

3. Determine the horizontal asymptote:

When the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

- The leading coefficient of the numerator ([tex]\(4x\)[/tex]) is 4.
- The leading coefficient of the denominator ([tex]\(x\)[/tex]) is 1.

Therefore, the horizontal asymptote [tex]\(y\)[/tex] is given by:
[tex]\[ y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} = \frac{4}{1} = 4 \][/tex]

So, the horizontal asymptote of the function [tex]\( y = \frac{4x + 32}{x - 8} \)[/tex] is [tex]\( y = 4 \)[/tex].
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