Limits help needed

[valid wagon]

I need help solving a limit: limit of sin(x)/x as x approaches 0.

[blissful arrow]

it should be 1

[twilit radish]

what are you allowed to use

[valid wagon]

how

[valid wagon]

trig identitities

[blissful arrow]

I think you can use the squeeze theorem

[blissful arrow]

ohhhh

[valid wagon]

how would we use squeeze theorem on that anyway

[blissful arrow]

we would use other functions 💀

[valid wagon]

i guess squeeze theorem would work

if i did it properly

[blissful arrow]

wait

[valid wagon]

bludicus

[blissful arrow]

my fault

yes you use the squeeze theorem

but you can only use trig identities

right?

[valid wagon]

that would be prefferable

[blissful arrow]

ok

[blissful arrow]

no these functions don't apply to the squeeze theorem

because it states that if f(x)<=g(x)<=h(x)

and f(p)=h(p)

[twilit radish]

also you can use 1 as the upper bound but you have to use that |sin x| ≤ |x| for all x

[blissful arrow]

then g(p) must also be equal to f(p) and h(p)

which means their limits are equal

[blissful arrow]

but in here

the function that is supposed to be less than sin(x)/x is greater at one point

so you will need a different function

[maiden lodge]

use taylor series expansion of sin

or squeeze sin(x) between x and x-x^3/6

[valid wagon]

what are you talking about

the limit of both functions as they approach 0 is 1

and their function values at 0 are 1

at other micro values around 0 and in general, it's either too big or too small

[blissful arrow]

the function that is supposed to be less than g(x)

which in this case is sin(x)/x

[blissful arrow]

it is greater than g(x) at some point

but it has to be

less than or equal to

or I may be wrong 🤷‍♂️

[valid wagon]

i remember it being equal to g(x) at one point but >= or <= at all the other points of the respective graphs

ngl I don't think it's possible w/o squeeze theorem

[twilit radish]

to be sure, are you only allowed to use squeeze theorem & trig identities? see if you can use l'hopital or taylor series

[maiden lodge]

Actually, it's solved historically way before l'hopitale's and it's various proofs include a lot of the tools used to prove l'hopitale

there is a geometric proof too

n-gons-->circles

[valid wagon]

update

it turns out you can use squeeze theorem

or just actually find the limit

by approximation

how do you end the question

[twilit radish]

.solved