Sandy asked in 科學數學 · 7 years ago

# 1題微積分的證明

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?

Update:

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• 7 years ago

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～

• Anonymous
7 years ago

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• 7 years ago

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7 years ago

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7 years ago

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7 years ago

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7 years ago

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7 years ago

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• 7 years ago

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.

• 7 years ago

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