How Do You Solve Challenging Limits and Analyze Sequence Monotonicity?

[xstetsonx]

Got a couple questions please help!

1. lim n>inf (x^4)/(e^(-2x))

How do you do this one? i know you use L hospital rule but you can never get rid of the denominator. or is there a trick i am missing?

2. Determine whether the sequences are increasing, decreasing, or not monotonic.
1.(cos(x))/(3^x)
2.(x-3)/(x+3)
3.1/(3x+6)

can you explain what to do with this one?

 

[LCKurtz]

Got a couple questions please help!

1. lim n>inf (x^4)/(e^(-2x))

How do you do this one? i know you use L hospital rule but you can never get rid of the denominator. or is there a trick i am missing?

You mean x --> infinity? Why would you use L'Hospital's rule? You have an infinity / 0 form.

2. Determine whether the sequences are increasing, decreasing, or not monotonic.
1.(cos(x))/(3^x)
2.(x-3)/(x+3)
3.1/(3x+6)

can you explain what to do with this one?

You could look at their derivatives.