can anybody help me w/ this?

Take the derivative of both the top and the bottom and substitute x=0

For example on number one.

derivative of sin(pi/6+x)-1/2 = cos(pi/6 + x)
derivative of x = 1

so limit = cos(pi/6 + 0)/1 = sqrt(3)/2

I leave the second one for you with this bit of additional help. You can check your work by putting a very small value of x into your calculator. If you solve this on any scientific calculator with x = .00001, you will see if your answer is correct.

that's the right answer -according to my teacher-
But we didn't get to deriviatives yet so I don't understand how you got the cos(pi/6 + x)

we're still doing limits, is there a way i can solve this problem using just limits? and maybe involve some trig?

or can u explain to me how to get the deriviatives of functions?

thanks a lot

Basically
If you traverse some further in your calculus textbook you will find necessary theorems put for solving such problems that my friend engineer has used to solve it

... but for your assistance i give you the first one solved using simple trig functions

We should know this

Sin(A+B) = Sin(A)Cos(B)+Cos(A)Sin(B)

Cos(A+B) = Cos(A)Cos(B) - Sin(A)Sin(B)


Lets start with the first one

Assume



Lest solve the numerator

sin(pi/6 + dx) - ½

= Sin(pi/6)Cos(dx) + Cos(pi/6)Sin(dx) - ½ ................... EQ(1)

Now here is the catch

Under Limits dx tends to zero
Sin(dx) also tends to 0 & Cos(dx) tends to 1
basics of limits say when x -> 0 then x is not equal to 0. So dx <> 0
So Sin(dx) -> 0 & Sin(dx) <> 0
Sin(dx) & dx both -> 0

Hence being very small they are nearly same

Sin(dx) = dx .........Remeber this
Cos(dx) = 1

Put these values in EQ(1)

We get

sin(pi/6 + dx) - ½
= Sin(pi/6)(1) + Cos(pi/6)(dx) - ½
= 0.5 + (Sqrt(3)/2)(dx) - 0.5
= (Sqrt(3)/2)(dx)

Putting it back in limits ...... we get



Under Limits as dx -> 0 ......... dx <> 0 ........... dx/dx = 1

So cancelling dx from numerator and denominator

L = Sqrt(3)/2 ............... Answer


Similary you can give a try for the Second one

Wow
Thanks guys...you're awesome!


*Edit--I don't get the catch


i'm like really bad in math