How do you show whether the improper integral #int lim (lnx) / x dx# converges or diverges from 1 to infinity?
Making the substitution, we have
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To determine whether the improper integral ( \int_{1}^{\infty} \frac{\ln(x)}{x} , dx ) converges or diverges, you can use the limit comparison test.

Select a Known Function: Choose a function that behaves similarly to ( \frac{\ln(x)}{x} ) near ( x = \infty ). A common choice is ( \frac{1}{x} ), as it has a similar behavior.

Formulate the Comparison: Consider the limit as ( x ) approaches infinity of the ratio of the two functions: [ \lim_{x \to \infty} \frac{\frac{\ln(x)}{x}}{\frac{1}{x}} ]

Simplify the Limit: Simplify the expression: [ \lim_{x \to \infty} \ln(x) = \infty ]

Apply the Limit Comparison Test: Since the limit is infinity, and the integral ( \int_{1}^{\infty} \frac{1}{x} , dx ) diverges (it's a known improper integral), by the limit comparison test, ( \int_{1}^{\infty} \frac{\ln(x)}{x} , dx ) also diverges.
Therefore, the improper integral ( \int_{1}^{\infty} \frac{\ln(x)}{x} , dx ) diverges.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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