Question

To the nearest .001,

lim (1-x)^(1/x)
x->0+

how to solve?

Answer 1

easy use the natural log.

e^ln((1-x)^(1/x))
now use the rules of ln :)
=
e^ln((1/x)ln(1-x))
=
(1/x)ln(1-x)

lim (1/x)ln(1-x)
x -> +0

are you allowed to use L'H rule?

Answer 2

what is the l'h rule?

Answer 3

l'hospital's rule

Answer 4

it is used to solve problems like these because you have 0 and infinity

Answer 5

we never used the natural log for this question though

Answer 6

well the answer to that question is .368
but what im wondering is how did it end up with that answer

Answer 7

are you going ahead of your class or just learning this for the hell of it?

Answer 8

this is what i learned in class
but i have no idea the process of it

Answer 9

do you understand the first steps?

Answer 10

of your method?

Answer 11

yes

Answer 12

I have to know this for a midterm so I don't mind helping you understand it infact I would be happy to help

Answer 13

thanks, and i understand that your useing th natural log in order to bring down the exponent

Answer 14

but how did the answer end up to .368?

Answer 15

right that is only the first step you need to use L'H rule to solve then you have to sub it back into the original problem

Answer 16

its a little more complex than what I have posted
if you notice I have infinity(0)

Answer 17

if you sub in 0+

Answer 18

I haven't gone through the method entirely do you want me to?

Answer 19

yes please do!

Answer 20

so we need to apply L'H rule which is used to solve problems such as these (remember to denote that you are using this rule or they will probably doc you)

I will do it on paper for you so give me some time

Answer 21

I will show you how to denote it

Answer 22

alright

Answer 23

to set up infinity/infinity or 0/0 we take one of the functions and divide it
so
f(x)g(x)
f(x) = infinity
g(x) = 0
so we can go f(x)(1/g(x)) for 0/0
or
g(x)/(1/f(x)) for infinity/infinity
then we take the derivative and simplify :)

Answer 24

usually it is best to go infinity/infinity tbh

Answer 25

if you dont know but it is usually pretty obvious which functions will derive nicely and which wont

Answer 26

alright, so what is the next step?

Answer 27

Im working on it

Answer 28

im doing it the wrong way first to show you the two choices

Answer 29

if thats ok

Answer 30

no problem

Answer 31

see that if I keep differentiating this it will always be 0/0 and thus it is the wrong choice

Answer 32

you can keep applying L'H rule over and over until you end up with a final value but only one choice will work to my knoweldge

Answer 33

I think I might of made a mistake deriving 1/x but yeah you see how that choice doesn't work I will actually solve it now

Answer 34

okayy

Answer 35

do you kind of see what I'm doing now though?

Answer 36

yeahh

Answer 37

there is one more step let me check my notes

Answer 38

yup :) I got this pellet
so we know that everything is equal to -1 so
e^(u) = the solution

-1 = u

Answer 39

e^(-1) = 0.368

Answer 40

do you follow how I solved that?

Answer 41

because ln((1-x)^(1/x)) = -1 we are just left with e^(-1)

Answer 42

ta da

Answer 43

at the 2nd equal. on top, how did you get the -1?

Answer 44

2nd equal sign. the -1 on top of the 1-x

Answer 45

take the derivative of ln(1-x) and then take the derivative of x

Answer 46

the derivative of ln(1-x) = x'/x
so whats the derivative of 1-x

Answer 47

its -1/1-x

Answer 48

ohhh okayy

Answer 49

then the -1

Answer 50

I hope you understand L'H rule now if you have any other questions ask away

Answer 51

yeah, is there an easier way to do this problem? i dotn remember being this long

Answer 52

Simplify always before moving on to the next derivative, unless it is too much of a mess to simplify

Answer 53

this is the only way to my knowledge and it really isn't that difficult if you have a firm grasp of derivatives and algebra. I recommend you practice some questions. here is a tip look at the two functions and put the cleanest derivative on the bottom

Answer 54

the reason I took so long is because I was explaining how to do this as well, and also I'm half asleep

Answer 55

trust me, your not the only one hahahha

Answer 56

Seriously though this pellet is easy try the question
lm x^(x^(2))
x->0+

Answer 57

or don't it is up to you I have the answer from my prof but if you get stuck at a part I can help you through it

Answer 58

okay hold on

Answer 59

is it always the natural log rule?

Answer 60

that is the only way I know how to solve this type of problem.

Answer 61

okay i dont know after i got x^2(ln x)

Answer 62

or did do that wrong

Answer 63

oh wait, now i find the dirivatives of both right?

Answer 64

this is a bad example lol you dont need to use L'H rule sorry about that. You can just solve it now lol

Answer 65

and e^(0) = 1

Answer 66

oh.....

Answer 67

I will give you a problem actually involving L'H rule this time I promise lol

Answer 68

lim (cos(x))^(1/(x^(2)))
x-> 0+

Answer 69

this is the problem that requires L'H rule sorry about that

Answer 70

oh god this looks complicated

Answer 71

its not too bad

Answer 72

just use the same method I did

Answer 73

okay. 1/(x^2)(ln cos x)?

Answer 74

now i find the derivative of the 1/x^2 and ln cos x

Answer 75

right because 1/0+ is infinity and ln(cos(0)) = ln(1) = 0

Answer 76

so now you have two functions g(x)f(x) here is where you apply the rule first look at the two functions which one will be the neatest to derive?

Answer 77

cox x?

Answer 78

no cos(x) is just going to be a repeated derivative and will never completely simplify so we don't want to put it as the denominator

Answer 79

ohhhh okay

Answer 80

1/infinity = 0 so we are going for 0/0

Answer 81

so put 1/x as the denominator

Answer 82

I would recommend writing it down exactly how I wrote it down.

Answer 83

how does it go?

Answer 84

not you have to apply the rule multiple times in this question and remember that tan(x) = sin(x)/cos(x)

Answer 85

sorry, its kinda hard doing it on the comp. without writing it down

Answer 86

write it down lol dont do it on the computer

Answer 87

you gave up fair enough Have a good night :)

Answer 88

the answer was 1/(e)^(1/2)