Question

i need to take the limit as x-->0 of 3x^2csc^2(x)...Is the answer 0?

Answer 1

Two things you need to know:

1. csc x = 1 / sin x

2. lim as x->0 (sin x / x) = 1

See if you can simplify your question using these equations, and arrive at an answer (which I think is not 0).

Answer 2

I am not sure how I am supposed to do it. does that mean the limit is 6?

Answer 3

Lim 3x^2 csc^2 can be written
x->0
Lim 3x^2/sin^2x
x->0
You can get the limit as x->0 using L'Hopital's rule: It is the ratio of the derivatives of numerator and denominator. You have to apply it twice here, since the first derivatives are also 0/0
Lim 3x^2/sin^2x
x->0
= Lim 6x/2sinxcosx
x->0
=Lim 6x/(sin 2x)
x->0
= 6/(2cos0) = 3

Answer 4

What Amrit suggested you do is to use the follwing rule:

Suppose Lim x ---> a of f(x) = y

Then Lim x ---> a of g(f(x)) = g(y)

if the function g(x) is continuous at the point x = y.

In this case you take f(x) to be
sin(x)/x and g(x) = 3 x^2. Because g(x) is a continuous function, the limit is 3*(1)^2 = 3.

Answer 5

Correction: f(x) = x/sin(x).