Question

find the limit as x approaches infinity f(x) = (e^-x)/x

Answer 1

\[f(x) = \frac{ e ^{-x} }{ x }\]

No idea x_x

Answer 2

greaaaat :(

Answer 3

Haha, I hate functions also. I might find it in my book, let's see....

Answer 4

What does e^-x look like as x goes to infinity?

Answer 5

exactly! it's part of our calc homework for the prerequisites and I'm honestly clueless

Answer 6

my calculator says it's unable to graph e^-x..

Answer 7

what does e^-x represent what is another way to write this?

Answer 8

I have functions hwk also :/

Answer 9

could it be -xln? I'm so confused

Answer 10

i understand. its ok

Answer 11

I'm clueless also although I'm intrigued...

Answer 12

e^-x=1/(e^x)

Answer 13

oh! so you can treat x as a regular negative exponent.. ok, I'm following, now what?

Answer 14

so, do you see that as x gets larger and increases without bound, so too does the denominator.

Answer 15

still there?

Answer 16

still here, if the denominator get's bigger, than the answer get's smaller yes?

Answer 17

you got it!

Answer 18

now remember you have product of x and e^x in the denominator, so it will get smaller real real fast.

Answer 19

notice the abundant use of technical terms here. lol

Answer 20

so the original x in the denominator stays there so it's 1/(x)(e^x)

Answer 21

exactly; so if you have a fraction that has a numerator of 1 and a denominator or zillions, what is the limit of this quotient?

Answer 22

0?

Answer 23

see? you are very smart. you got the answer yourself. 0 is correct. Nicely done. Anything else i can help you with

Answer 24

the back of the book says it's 1 though....

Answer 25

either the problem is incorrect or the answer is incorrect.

Answer 26

are you familiar with L'Hopitals rule?

Answer 27

it says to use tables and graphs to find the answers.. find the limit as x approaches infinity when f(x) = e^-x/x then find the limit as x approaches f(x) from the left, then find any horizontal asymptotes if they exsist..... fml

Answer 28

never heard of it, why?

Answer 29

ok. we just found the horizontal asymptote

Answer 30

no you were right! i was looking at the wrong answers I think.. so confused

Answer 31

if you want a vertical asymptote, then you have to look at the quotient around x=0

Answer 32

it is ok. now, realize that you did most of this problem yourself. i kept nudging you in the right direction.
have some faith in your own abilities

Answer 33

can i do anything else for you? Further explanation? any other questions?

Answer 34

thank you so much! hold on one moment,trying to record the work for this one, and then analyze the rest of the question

Answer 35

take your time. let me know if you have any questions as you record.

Answer 36

so that makes sense, but what about when it's approaching negative infinity? how could I figure that out?

Answer 37

ok. what does e^-x look like as x->\[-\infty\]?

Answer 38

do you see that the exponent increases positively without bound?

Answer 39

so it's infinity? but what about the x in the denominator? do I just ignore it?

Answer 40

no, never ignore it. But, we know that exponential functions increase much more rapidly than linear functions. Do you see that?

Answer 41

so why are we not including it? and only looking at e^-x?

Answer 42

of course, they increase exponentially ;)

Answer 43

because the exponential function gets so large so fast; think about it
and take e^200 and divide by -200.. oops. from the left it will turn that quotient negative though, so the graph decreases without boutnd

Answer 44

bound*

Answer 45

does this make sense?

Answer 46

so multiplying by that extra x is just making it decrease even faster?

Answer 47

no, but because it divides a huge pos number, it just barely slows it down

Answer 48

you are on the right track here.

Answer 49

remember that the exponent is the neg of a negative

Answer 50

Plugging the equation into my calculator is just confusing me even more though, but that's how our teacher want's us to be able to prove it by graph.... uh.... head hurts

Answer 51

what does the graph look like on your calculator?

Answer 52

I think you only have to graph the function for large numbers to solve the above problem, dont u?