Conclude that ln x can be made as large as desired by choosing x sufficiently large
What does this imply about lim x→∞ ln x?
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2014-04-29 6:09 am
Use the fact that ln 4 > 1 to show that ln 4^m>m for m>1
Conclude that ln x can be made as large as desired by choosing x sufficiently large
What does this imply about lim x→∞ ln x?
Conclude that ln x can be made as large as desired by choosing x sufficiently large
What does this imply about lim x→∞ ln x?
回答 (11)
2014-04-30 5:49 am
✔ 最佳答案
ln 4 > 1 m ln 4 > m
ln 4^m > m
2014-04-28 23:19:46 補充：
When x is large, ln 4^x is larger, ln 4^{ln 4^x} is even larger...
This implies that lim x→∞ ln x does not exist (or the limit is +∞)
2014-04-29 21:49:29 補充：
Thank you Sandy~ (◕‿◕✿)
As discussed in the comment box,
consider
ln 4 > 1
m ln 4 > m [by multiplying m > 1 to both sides]
ln 4^m > m [by property of logarithm]
Therefore, for any m > 1, ln 4^m would be greater than m.
When x is large, ln 4^x is larger, ln 4^{ln 4^x} is even larger...
This implies that lim x→∞ ln x does not exist (or the limit is +∞).
2014-04-30 14:39:17 補充：
謝謝 老怪物 的意見～
希望發問者明白～
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2014-04-30 6:26 pm
關於最後一個小問題 x→+∞ 時 ln(x) 的行為.
這需要一個結果 (定理): ln(x) 是 strictly increasing 的.
因此, 當 x 可以任意大時, 可找一個上升數列 m_1, m_2,...
x 大於 m_1, 所以 ln(x) > ln(m_1);
x 大於 m_2, 所以 ln(x) > ln(m_2);
以此類推.
所以 x 無限增大時, ln(x) 也無限增大.
這需要一個結果 (定理): ln(x) 是 strictly increasing 的.
因此, 當 x 可以任意大時, 可找一個上升數列 m_1, m_2,...
x 大於 m_1, 所以 ln(x) > ln(m_1);
x 大於 m_2, 所以 ln(x) > ln(m_2);
以此類推.
所以 x 無限增大時, ln(x) 也無限增大.
2014-04-29 9:22 am
收錄日期: 2021-05-04 01:55:17
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20140428000015KK07357