Find the limit of (e^-3x)(cos9x) when x--->infinity. State whether the limit is (some numbers), negative infinity/positive infinity, does not exist, or use any possible ways to find the limit.

My calculations:

Method1:

I put it in the calculator and used the data table to replace x with any positive numbers since x--> +infinity. The answer is -infinity, since x replace with any positive numbers the result will always came out negative.

Method2:

Get the equation out of the limit and make it an equation: y = (e^-3x)(cos9x)

Then I used the derivative of it, and since it multiplies I have to use the Product Rules.

y = (-3e^-3x)(cos9x) + (e^-3x)(-9sin9x) = 3e^-3x(-cos9x - 3sin9x)

It is not the answer but my teach would accept an equation written out with limit.

Am I approaching this equation correctly?