Question

Show that the limit as x->0 of 3x^2/(\sin(4x^2))=3/4.
Thanks. (Without l'Hospital's)

Answer 1

\[Lim(x \rightarrow 0)\frac{3x^2}{(\sin(4x^2)}\]

Answer 2

Use\[Lim (x \rightarrow 0) \frac{sinx}{x}=1\rightarrow sinx=x\]

Answer 3

Why is this true? It's obvious if you graph it http://fooplot.com/#W3sidHlwZSI6MCwiZXEiOiJzaW4oeCkiLCJjb2xvciI6IiMwMDAwMDAifSx7InR5cGUiOjAsImVxIjoieCIsImNvbG9yIjoiIzAwMDAwMCJ9LHsidHlwZSI6MTAwMH1d
sinx=x when you move closer to the origin.

Answer 4

Yes, but, remember that \[nx\] is not of the form \[nx^2\] so the equivalent does not necessarily apply. :( Now, were we to show that \[\sin nx^2 \sim x^2, x \to 0\] that would make more sense... How would we ago about doing that, though?

Answer 5

\[ \lim_{x \rightarrow 0}\frac{4x^2}{\sin(4x^2)} \times {3 \over 4} \\
\lim_{4x^2 \rightarrow 0}\frac{4x^2}{\sin(4x^2)} \times {3 \over 4} = {3 \over 4}\]

Answer 6

\[ \lim_{4x^2 \rightarrow 0}\frac{1}{\frac{\sin(4x^2)}{4x^2}} \times {3 \over 4} = {3 \over 4} \]

Answer 7

nice piece of app
http://fooplot.com/#W3sidHlwZSI6MCwiZXEiOiJzaW4oeCkiLCJjb2xvciI6IiMwMDAwMDAifSx7InR5cGUiOjAsImVxIjoieCIsImNvbG9yIjoiIzAwMDAwMCJ9LHsidHlwZSI6MTAwMCwid2luZG93IjpbIi05LjEyIiwiMy44Nzk5OTk5OTk5OTk5OTg2IiwiLTEuOTQwMDAwMDAwMDAwMDAwNCIsIjYuMDYwMDAwMDAwMDAwMDAwNSJdfV0-

Answer 8

Or\[\lim(x \rightarrow 0) \frac{3x^2}{\sin(4x^2)}=\lim(x \rightarrow 0) \frac{3x^2}{4x^2}=\lim(x \rightarrow 0) \frac{3}{4}\]

Answer 9

Ahh, yes, again, although I don't know how to prove they're asymptotically equal, I should be able to solve it, now, thanks.