Originally Posted by

**Mathstud28** Today in one of my texts I saw this

$\displaystyle \text{As}x\to{0},\tan(x^3)\sim{x^2}$

I dont think this is right?

Correct me if I am wrong but Three ways prove I am right unless I am missing somethign stupid

$\displaystyle \lim_{x\to{0}}\frac{\tan(x^3)}{x^2}=\lim_{x\to{0}} \frac{\frac{\sin(x^3)}{\cos(x^3)}}{x^2}=\lim_{x\to {0}}\frac{sin(x^3)}{\cos(x^3)\cdot{x^2}}$

Now I will apply that $\displaystyle \sin(x^3)\sim{x^3}\text{As}x\to{0}$

To rewrite this as $\displaystyle \lim_{x\to{0}}\frac{x^3}{\cos(x^3)\cdot{x^2}}=\lim _{x\to{0}}\frac{x}{\cos(x^3)}=0\ne{1}$

You can also do this limit through power series and L'hopital's.

Where am I making my mistake?