limit problem involving sin please help

[HMezz]

as x approaches zero, sin8x/x^2

I thought it was infinity but the answer was marked incorrect. Does the limit not exist?

 

[Ishuda]

as x approaches zero, sin8x/x^2

I thought it was infinity but the answer was marked incorrect. Does the limit not exist?

\(\displaystyle \lim_{x\to0}\frac{sin(8x)}{x^2} = \infty\)
and, since \(\displaystyle \infty\) is not a real number, one could say the limit does not exist (as a real number). However
\(\displaystyle \lim_{x\to0}(\frac{sin(8x)}{x})^2\)
does exist.

EDIT:Thanks pka. Fix limit; not to infinity but to zero.

 

[pka]

as x approaches zero, sin8x/x^2

\(\displaystyle \displaystyle{\lim _{x \to 0}}\frac{{\sin (8x)}}{{{x^2}}} = {\lim _{x \to 0}}\frac{{\sin (8x)}}{{8x}} \cdot {\lim _{x \to 0}}\frac{8}{x} = 1 \cdot {\lim _{x \to 0}}\frac{8}{x}\)

\(\displaystyle \displaystyle{\lim _{x \to 0}}\frac{8}{x}\) does not exist.