MHB Limit of a trigonometric function

[Yankel]

Hello all,

I need some guidance in solving these limits:

\[\lim_{x\rightarrow \infty }x\cdot sin(x)\]

\[\lim_{x\rightarrow 0 }\frac{sin(x)}{\sqrt{x}}\]

\[\lim_{x\rightarrow \infty }\frac{sin(x)}{x}\]

I guess that the second and third ones are somehow related to

\[\lim_{x\rightarrow 0 }\frac{sin(x)}{x}\]

But I am not sure how to convert the known limit into the new ones.

Thank you in advance.

 

[Prove It]

Hello all,

I need some guidance in solving these limits:

\[\lim_{x\rightarrow \infty }x\cdot sin(x)\]

\[\lim_{x\rightarrow 0 }\frac{sin(x)}{\sqrt{x}}\]

\[\lim_{x\rightarrow \infty }\frac{sin(x)}{x}\]

I guess that the second and third ones are somehow related to

\[\lim_{x\rightarrow 0 }\frac{sin(x)}{x}\]

But I am not sure how to convert the known limit into the new ones.

Thank you in advance.

Hint for the third: The product of a function that goes to 0 with a bounded function must have a limit of 0.

 

[Yankel]

I see. Can I use the same logic in the first as well ?

 

[Prove It]

I see. Can I use the same logic in the first as well ?

No, neither function goes to 0.

As for whether the function has an infinite limit, as it oscillates I would lean towards no...