Calculate the following limit: lim -x/tan(3x) as x approches to 0

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justaguide eNotes educator| Certified Educator

You can only ask one question at a time. I have edited the question to comply with the same.

We have to find the value of lim x--> 0 [ -x/tan(3x)]

If we substitute x = 0, we get the indeterminate form 0/0. This allows the use of l'Hopital's rule. We substitute the numerator and denominator by their limits.

lim x--> 0 [ -x/tan(3x)]

=> lim x--> 0[ -1/3*(sec 3x)^2]

substitute x = 0

=> -1/3

The required limit is -1/3