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Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) ln(5n) ln(15n) lim n→∞ ln(5n) ln(15n)

Respuesta :

LammettHash

If the limand is [tex]\ln(5n)\ln(15n)[/tex], then the limit diverges. But if you mean [tex]\dfrac{\ln(5n)}{\ln(15n)}[/tex], we can do some manipulating to rewrite it as

[tex]\displaystyle\lim_{n\to\infty}\frac{\ln(5n)}{\ln(15n)}=\lim_{n\to\infty}\frac{\ln5+\ln n}{\ln15+\ln n}=\lim_{n\to\infty}\frac{\frac{\ln5}{\ln n}+1}{\frac{\ln15}{\ln n}+1}[/tex]

[tex]\ln n\to\infty[/tex] as [tex]n\to\infty[/tex], so the fractional terms vanish and you're left with 1.

Cricetus

The limit will be "1". A complete solution is provided below.

According to the question,

By using the L'Hopital's rule, we get

→ [tex]\lim_{x \to \infty} {\frac{ln(5n)}{ln(15n)} } = \lim_{x \to \infty} {\frac{(\frac{1}{5n} )(5)}{(\frac{1}{15n} )(15)} }[/tex]

→                          [tex]= \lim_{x \to \infty} {\frac{(\frac{1}{n} )}{(\frac{1}{n} )} }[/tex]

→                          [tex]= \lim_{x \to \infty} 1[/tex]

→                          [tex]= 1[/tex]

Thus the above answer is correct.

Learn more about Limits here:

https://brainly.com/question/2264931

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