To find the horizontal asymptote of the function [tex]\( g(x) = \frac{21 x^2}{7 x^2 + 4} \)[/tex], we need to consider the behavior of the function as [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]).
1. Analyze the degrees of the polynomial:
The function [tex]\( g(x) \)[/tex] is a rational function, which is the ratio of two polynomials. The degree of the polynomial in the numerator (top part) is 2, because the highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex]. Similarly, the degree of the polynomial in the denominator (bottom part) is also 2, as the highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex].
2. Horizontal asymptote rule for rational functions:
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (the function diverges to infinity or minus infinity).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and the denominator.
In our function [tex]\( g(x) = \frac{21 x^2}{7 x^2 + 4} \)[/tex], the degrees of the numerator and denominator are equal, as both are 2.
3. Identify the leading coefficients:
- The leading coefficient of the numerator is 21 (from [tex]\( 21 x^2 \)[/tex]).
- The leading coefficient of the denominator is 7 (from [tex]\( 7 x^2 \)[/tex]).
4. Calculate the horizontal asymptote:
The horizontal asymptote is the ratio of the leading coefficients:
[tex]\[
y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}} = \frac{21}{7} = 3
\][/tex]
Therefore, the horizontal asymptote of the function [tex]\( g(x) = \frac{21 x^2}{7 x^2 + 4} \)[/tex] is [tex]\( y = 3 \)[/tex].
