Question

By graphing y=(x+1)^1/x, estimate
lim y =(1+x)^1/x
x->0

*round to one decimal place

how do i determine this?

Answer 1

logarithm

Answer 2

how can I start that?

would it be lim y= lim 1/x (1+x) ?
x->0

Answer 3

ln(y) = (1/x) ln(x+1)

Answer 4

then use L'hospital rule

Answer 5

oops i put lim when i meant ln haha

okay... could you please refresh me on what the L'hospital rule is? is that when it equals zero?

Answer 6

I think they want you to graph it.

Answer 7

ln(x+1)/x , applying L'hospital rule, d/dx ln(x+1) / d/dx(x)

Answer 8

oh like would i put y=(x+1)^1/x into wolfram?

Answer 9

lol, yeah they want you to graph it. Why go through all the trouble XD

Answer 10

yes, that would work

Answer 11

http://www.wolframalpha.com/input/?i=+y%3D%28x%2B1%29%5E1%2Fx

is it the top or bottom graph?

Answer 12

like this

Answer 13

both are correct. But the BLUE line is the graph.

Answer 14

oh okay :)

and @phi we need both graphs?

Answer 15

that you attached i mean :P

Answer 16

wolfram is showing a "zoomed" in version in the top one (but not much of a zoom)

Answer 17

I used geogebra to plot the function.

If you want a good estimate of f(0), you have to zoom in...

Answer 18

http://www.wolframalpha.com/input/?i=+y%3D%28x%2B1%29%5E%281%2Fx%29%2C+x+%3D+0.000001

limit is actually e

Answer 19

ohh okay...

so would it be 2.719?


so about 2.7 rounded to one decimal place?

Answer 20

yes... I zoomed in a bit more than needed. as you can see, it is "e"

Answer 21

more or less, yes

Answer 22

okay, and e=2.7?

Answer 23

no, e is an irrational number. But approximately 2.8

Answer 24

at least in this case?

Answer 25

ohh okay thanks you guys!!

Answer 26

oops i mean approximately 2.7 but yea, in this case, you can say 2.7

Answer 27

haha okie :)thanks!

Answer 28

if interested, see
http://www.khanacademy.org/economics-finance-domain/core-finance/interest-tutorial/cont-comp-int-and-e/v/e-as-limit

Answer 29

thanks for the link! it was a great video! So it looks like the value of e is basically 2.718.... haha... it's like pi all over again!! hahaa :p