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

1. henpen

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

2. henpen

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

3. henpen

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

4. LolWolf

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?

5. experimentX

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

6. experimentX

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

7. experimentX
8. henpen

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

9. LolWolf

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