limits: lim (1/sin(x))-(1/x), x>infinity

[ssabadnama3]

lim (1/sin(x))-(1/x) all the directions said was find the limit i tried to do the limit from the positive and negative side but i didn't get far
x>infinity

 

[galactus]

Re: limits

\(\displaystyle \lim_{x\to \infty}\left[\frac{1}{sin(x)}-\frac{1}{x}\right]=\lim_{x\to \infty}\frac{1}{sin(x)}-\lim_{x\to \infty}\frac{1}{x}\)

The rightmost limit is obviously 0.

Look at 1/sin(x)=csc(x). What can you say about that as x gets larger and larger?.

 

[ssabadnama3]

im not sure what csc does as x gets larger. i know what the graph of csc is. is it where it goes off into infinity?

 

[galactus]

csc(x) does not converge. The limit is undefined.

 

[mark]

think about the graph of sinx. what does the graph do? what happens at sin x at 0,180,360,etc

 

[BigGlenntheHeavy]

This is known as divergence by oscillation, as x goes to infinity, csc(x) oscillates between - infinity and infinity.