Question

Let f(x) =cos x/e^x= e^-x*cos x
Prove that f has a horizontal asymptote as x -> +infinity, and find that
asymptote.

Answer 1

does using squeeze theorem work?

Answer 2

i did us the squeeze theorem. but i don't know what the limit of cos x/ e^x does.
I got 0 for the limit. then what does it tell us?

Answer 3

if the limx->infinity is 0, then the horizontal asymtote should be at y=0 i think

Answer 4

i tried to find how the graph looks like. Try www.wolframalpha.com. and input 'horizontal asymptote f(x)=cosx/e^x' There's no horizontal asymptote shown when y=0.

Answer 5

As you know, cos x oscillates between -1 and 1.

So multiplying it by e^-x basically squeezes that envelope down so cos is now oscillating between e^-x and -e^-x

For example in this graph (I've 'speeded up' the oscillating of the cos function here so you can see this more clearly).

Answer 6

hey veryconfused what question you on?

Answer 7

Now clearly this envelope between -e^-x and e^-x in the limit reduces down to zero. Hence the graph of cos x . e^-x also approaches the x-axis as x--> infinity.

Thus in fact, y = 0 -- i.e., the x-axis -- is an asymptote of the graph of this function.

Answer 8

i am now on questino 5

Answer 9

ya eh -this is my last q other than the maple that i cant get to work

Answer 10

but why if we look at the graph, it's crossing the x axis?
asymptotes are not supposed to be touched by the graph, aren't they?

Answer 11

i didnt even start on the lab yet, does it take long?

Answer 12

no but i cant get mine workin