# How do you find the limit of #[ln(6x+10)-ln(7+3x)]# as x approaches infinity?

Convert to a fraction and then use L'Hôpital's rule. Answer:

Compute the derivative of numerator:

Compute the derivative of the denominator:

Make a new fraction:

Therefore, the original limit goes to the same value:

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To find the limit of [ln(6x+10)-ln(7+3x)] as x approaches infinity, we can simplify the expression using logarithmic properties. By applying the quotient rule of logarithms, we can rewrite the expression as ln((6x+10)/(7+3x)).

As x approaches infinity, the terms with the highest degree (6x and 3x) dominate the expression. Therefore, we can ignore the constants (10 and 7) and simplify the expression further to ln(6x/3x).

Simplifying this gives us ln(2), which is a constant. Thus, the limit of [ln(6x+10)-ln(7+3x)] as x approaches infinity is ln(2).

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