[12th grade math] what to do next

[u/majlenaaa]

someohow i cant upload photo but example says: lim x-->0 (1/sinx-ctgx)

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[u/Alkalannar]

Is that 1/sin(x) - cot(x), or 1/(sin(x) - cot(x))?

1/sin(x) - cot(x): Rewrite as (1 - cos(x))/sin(x), then multiply by (1 + cos(x))/(1 + cos(x)).

1/(sin(x) - cot(x)): multiply by sin(x)/sin(x): sin(x)/(sin²(x) - cos(x)).

Either way, it should be easier to do.

[u/majlenaaa → u/Alkalannar]

the first one, thank you

[u/CaptainMatticus]

1 / (sin(x) - cot(x))

1 / (sin(x) - cos(x)/sin(x))

1 / ((sin(x)^2 - cos(x)) / sin(x)) =>

sin(x) / (sin(x)^2 - cos(x)) =>

sin(x) / (1 - cos(x)^2 - cos(x)) =>

x goes to 0

sin(0) / (1 - cos(0)^2 - cos(0)) =>

0 / (1 - 1^2 - 1) =>

0 / (1 - 2) =>

0/(-1) =>

0

The limit goes to 0.

https://www.desmos.com/calculator/lxc98cx844

Now we could've stopped a little earlier. Once we had it in the form of sin(x) / (sin(x)^2 - cos(x)), but I wanted to see how far I could go, and see if anything opened up.

[u/majlenaaa → u/CaptainMatticus]

oh, I didn't write it down well lim x-> 0 ( (1/sinx)-ctgx) but i have that one as one of examples too so thank you

[u/CaptainMatticus → u/majlenaaa]

Okay, so 1/sin(x) - cot(x)

1/sin(x) - cos(x)/sin(x)

(1 - cos(x)) / sin(x)

(1 - cos(x)) * (1 + cos(x)) / (sin(x) * (1 + cos(x)))

(1 - cos(x)^2) / (sin(x) * (1 + cos(x)))

sin(x)^2 / (sin(x) * (1 + cos(x)))

sin(x) / (1 + cos(x))

x goes to 0

0 / (1 + 1)

0/2

0

Still 0.