Warren Wells
07/06/2024 · Senior High School
b. \( \lim _{n \rightarrow \infty} \frac{n^{2}-5 n-1}{9 n^{2}+2} \)
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Tutor-Verified Answer
Step-by-step Solution
Calculate the limit \( \lim _{n \rightarrow \infty} \frac{n^{2}-5 n-1}{9 n^{2}+2} \).
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
\(\lim _{n\rightarrow +\infty}\left(\frac{n^{2}-5n-1}{9n^{2}+2}\right)\)
- step1: Rewrite the expression:
\(\lim _{n\rightarrow +\infty}\left(\frac{n^{2}\left(1-\frac{5}{n}-\frac{1}{n^{2}}\right)}{n^{2}\left(9+\frac{2}{n^{2}}\right)}\right)\)
- step2: Reduce the fraction:
\(\lim _{n\rightarrow +\infty}\left(\frac{1-\frac{5}{n}-\frac{1}{n^{2}}}{9+\frac{2}{n^{2}}}\right)\)
- step3: Rewrite the expression:
\(\frac{\lim _{n\rightarrow +\infty}\left(1-\frac{5}{n}-\frac{1}{n^{2}}\right)}{\lim _{n\rightarrow +\infty}\left(9+\frac{2}{n^{2}}\right)}\)
- step4: Calculate:
\(\frac{1}{\lim _{n\rightarrow +\infty}\left(9+\frac{2}{n^{2}}\right)}\)
- step5: Calculate:
\(\frac{1}{9}\)
La expresión \( \lim _{n \rightarrow \infty} \frac{n^{2}-5 n-1}{9 n^{2}+2} \) se puede simplificar como \( \frac{1}{9} \) o como \( 0.\dot{1} \).
Quick Answer
La expresión se simplifica a \( \frac{1}{9} \) o \( 0.\dot{1} \).
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