Answer :

To find the limit [tex]\(\lim _{x \rightarrow \infty} \frac{9 \cdot x^7-x^2}{5 \cdot x-9 \cdot x^3+2 \cdot x^5}\)[/tex], we need to analyze the behavior of the numerator and the denominator as [tex]\(x\)[/tex] approaches infinity.

1. Identify the leading term of each polynomial:
- In the numerator [tex]\(9 \cdot x^7 - x^2\)[/tex], the leading term is [tex]\(9 \cdot x^7\)[/tex] because it has the highest power of [tex]\(x\)[/tex].
- In the denominator [tex]\(5 \cdot x - 9 \cdot x^3 + 2 \cdot x^5\)[/tex], the leading term is [tex]\(2 \cdot x^5\)[/tex] because it has the highest power of [tex]\(x\)[/tex].

2. Focus on these leading terms:
- As [tex]\(x\)[/tex] approaches infinity, the lower degree terms ([tex]\(-x^2\)[/tex] in the numerator and [tex]\(5x - 9x^3\)[/tex] in the denominator) become insignificant compared to the leading terms. Therefore, the expression can be simplified to:
[tex]\[
\frac{9 \cdot x^7}{2 \cdot x^5}
\][/tex]

3. Simplify the fraction:
- Simplify the expression by dividing both the numerator and the denominator by [tex]\(x^5\)[/tex]:
[tex]\[
\frac{9 \cdot x^7}{2 \cdot x^5} = \frac{9 \cdot x^{7-5}}{2} = \frac{9 \cdot x^2}{2}
\][/tex]

4. Analyze the limit:
- Now, find the limit as [tex]\(x\)[/tex] approaches infinity:
[tex]\[
\lim_{x \to \infty} \frac{9 \cdot x^2}{2}
\][/tex]
- As [tex]\(x\)[/tex] grows larger, [tex]\(x^2\)[/tex] also increases without bound, meaning the expression [tex]\(\frac{9 \cdot x^2}{2}\)[/tex] becomes infinitely large.

Therefore, the limit is [tex]\(\infty\)[/tex], often written as [tex]\(\infty\)[/tex].

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