l'hospital's rule, help with limit pls as x approaches inf (14x/[14x+7])^3x?

By "multiplied the top and bottom by Ln" I assume you mean you took the natural log of both. Ln is an operation, not a variable, so you're not "multiplying by Ln".

In any case, you can't necessarily take the log (of any base) of both the top and bottom of a fraction. For example 1/2 is not Ln(1) / Ln(2), because that would be 0 / Ln(2) = 0.

You can, however, use the identity Y = e^(Ln Y). This gives you the denominator of e^[ Ln((14x+7)^3x)] = e^[3x Ln(14x+7)]. The deriviative of this is just e^[3x Ln(14x+7)] *( 3Ln(14x+7) + 3x( 14/(14x+7) )). Note that you can change 3x( 14/(14x+7)) to 42x / (14x+7) = 42 / (14 + (7/x)). So the entire expression on the bottom goes to infinity, while the derivative of the top (14x) is just 14. So the limit goes to zero.

You cant "mutliply by ln". What you can do is this: f(x)= e^ln f(x)

or f(x) = ln e^f(x)

Then you can do what you want to do, and, finally, you may apply l'Hopital

Ana