To find the limit [tex]\(\lim_{x \rightarrow \infty} \frac{8x^2 + 7x - 10}{2x^3 - 3x - 4}\)[/tex], let's analyze the behavior of the function as [tex]\(x\)[/tex] approaches infinity.
1. Identify the degrees of the polynomials:
- The degree of the numerator [tex]\(8x^2 + 7x - 10\)[/tex] is 2.
- The degree of the denominator [tex]\(2x^3 - 3x - 4\)[/tex] is 3.
2. Compare the degrees:
- The degree of the numerator (which is 2) is less than the degree of the denominator (which is 3).
3. Understand the implications:
- When the degree of the numerator is less than the degree of the denominator, the limit of the rational function as [tex]\(x\)[/tex] approaches infinity is 0. This is because the higher degree term in the denominator will grow much faster than the highest degree term in the numerator as [tex]\(x\)[/tex] becomes very large.
4. Conclude the result:
- Therefore, we can conclude that:
[tex]\[
\lim_{x \rightarrow \infty} \frac{8x^2 + 7x - 10}{2x^3 - 3x - 4} = 0
\][/tex]
So, the result of the limit is [tex]\(0\)[/tex].
