Question

Limits,


can somebody tell me how would you solve the limit n * sin(pi/n) as n goes to infinity

Answer 1

Maybe start by substituting pi/n = x

Answer 2

As n goes to infinity, where will x go to ?

Answer 3

infinity

Answer 4

Think a bit

Answer 5

sorry I ment zero

Answer 6

Good. Just notice that the sequence
1/2, 1/3, 1/4, 1/5, ...

approaches 0 as the bottom gets large

Answer 7

still we have infinity * ero

Answer 8

zero *

Answer 9

Yes, what is the expression after making the substitution ?

Answer 10

n(sin(x)) ?

Answer 11

Don't want to see any n's

Answer 12

pi/x * sin(x)

Answer 13

uhmmm

Answer 14

this feels weird

Answer 15

please show the limit also

Answer 16

yeah it should feel weird

Answer 17

is the limit zero

Answer 18

I dont think I have a denominator after the sub

Answer 19

so we have to figure out the limit sin(x)/x as x tends to 0

Answer 20

scratch that

Answer 21

so basically this is 0/0

Answer 22

Yes, you may use Lhospital if you want

Answer 23

but I dont have inf/inf

Answer 24

or does LHS apply with 0/0 too

Answer 25

Yes it applies to both 0/0 and inf/inf

Answer 26

aha!

Answer 27

thanks a lot

Answer 28

There are few other ways to do this. I'd like to show you another way

Answer 29

Ok

Answer 30

Remember the limit definition of derivative ?

Answer 31

yup

Answer 32

How do you find the derivative of sinx at x = 0 ?

Answer 33

lim y-> inf (f(x +y) -f(x) / y

Answer 34

) *

Answer 35

0 * *

Answer 36

\[[f(x)]'(a) = \lim\limits_{x\to a}~\dfrac{f(x) - f(a)}{x-a}\]

Answer 37

Plugin f(x) = sinx and a = 0

Answer 38

lim x -> 0 of [sin(x) - sin(0)] / x

Answer 39

so basically LHS ?

Answer 40

because that leaves me with a 0/0 form once again

Answer 41

lim x -> 0 of sin(x) / x

Answer 42

~~LHS~~

Answer 43

hm

Answer 44

im getting a bit rusty :D

Answer 45

lim x->0 cos(x)

Answer 46

I guess using LHS with the definition of the derivative is like a no go

Answer 47

:D

Answer 48

Hey still here ?

Answer 49

The idea here is to use the derivative to figure out the limit value

Answer 50

The derivative of sinx is cosx
so the limit evaluates to cos(0)

Answer 51

yes

Answer 52

so lim x->0 cos(x) = 1