Answer :
To evaluate the limits of the rational function [tex]\( f(x) = \frac{12x^6 - 8}{4x^6 - 7x^5} \)[/tex] as [tex]\( x \)[/tex] approaches infinity and negative infinity, and to find any horizontal asymptotes, follow these steps:
1. Identify the Highest Degree Terms:
- In the numerator, the highest degree term is [tex]\( 12x^6 \)[/tex].
- In the denominator, the highest degree term is [tex]\( 4x^6 \)[/tex].
2. Evaluate the Limit as [tex]\( x \to \infty \)[/tex]:
- When dealing with limits at infinity for rational functions, focus on the highest degree terms because they dominate the behavior of the function.
- The function simplifies to:
[tex]\[
f(x) = \frac{12x^6}{4x^6} = \frac{12}{4} = 3
\][/tex]
- Therefore, [tex]\(\lim_{x \to \infty} f(x) = 3\)[/tex].
3. Evaluate the Limit as [tex]\( x \to -\infty \)[/tex]:
- When [tex]\( x \to -\infty \)[/tex], the highest degree terms still dominate.
- The simplified fraction is the same:
[tex]\[
f(x) = \frac{12x^6}{4x^6} = \frac{12}{4} = 3
\][/tex]
- Thus, [tex]\(\lim_{x \to -\infty} f(x) = 3\)[/tex].
4. Determine the Horizontal Asymptote:
- If the limits as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex] are both finite, that value is a horizontal asymptote.
- Since both limits as [tex]\( x \)[/tex] approaches infinity and negative infinity are 3, the horizontal asymptote is [tex]\( y = 3 \)[/tex].
So, we find that:
- [tex]\(\lim_{x \to \infty} f(x) = 3\)[/tex],
- [tex]\(\lim_{x \to -\infty} f(x) = 3\)[/tex],
- The horizontal asymptote of [tex]\( f(x) \)[/tex] is [tex]\( y = 3 \)[/tex].
1. Identify the Highest Degree Terms:
- In the numerator, the highest degree term is [tex]\( 12x^6 \)[/tex].
- In the denominator, the highest degree term is [tex]\( 4x^6 \)[/tex].
2. Evaluate the Limit as [tex]\( x \to \infty \)[/tex]:
- When dealing with limits at infinity for rational functions, focus on the highest degree terms because they dominate the behavior of the function.
- The function simplifies to:
[tex]\[
f(x) = \frac{12x^6}{4x^6} = \frac{12}{4} = 3
\][/tex]
- Therefore, [tex]\(\lim_{x \to \infty} f(x) = 3\)[/tex].
3. Evaluate the Limit as [tex]\( x \to -\infty \)[/tex]:
- When [tex]\( x \to -\infty \)[/tex], the highest degree terms still dominate.
- The simplified fraction is the same:
[tex]\[
f(x) = \frac{12x^6}{4x^6} = \frac{12}{4} = 3
\][/tex]
- Thus, [tex]\(\lim_{x \to -\infty} f(x) = 3\)[/tex].
4. Determine the Horizontal Asymptote:
- If the limits as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex] are both finite, that value is a horizontal asymptote.
- Since both limits as [tex]\( x \)[/tex] approaches infinity and negative infinity are 3, the horizontal asymptote is [tex]\( y = 3 \)[/tex].
So, we find that:
- [tex]\(\lim_{x \to \infty} f(x) = 3\)[/tex],
- [tex]\(\lim_{x \to -\infty} f(x) = 3\)[/tex],
- The horizontal asymptote of [tex]\( f(x) \)[/tex] is [tex]\( y = 3 \)[/tex].
