<s>[INST] Doubt about asymptotic analysis $\frac{1}{x^{a}*(4+9x)^{b+1}}\sim \frac{1}{9^{b+1}x^{a+b+1}}$ I don't understand why is correct
$$a,b\in \mathbb{R}$$
$$x \mapsto \infty $$
$$\frac{1}{x^{a}*(4+9x)^{b+1}}\sim \frac{1}{9^{b+1}x^{a+b+1}}$$
I would write
$$x \mapsto \infty $$
$$\frac{1}{(4+9x)^{b+1}}\sim \frac{1}{9x^{b+1}}$$
so 
$$\frac{1}{x^{a}*(4+9x)^{b+1}}\sim \frac{1}{9x^{a+b+1}}$$
Where I made a mistake?
Someone can help me?
Thanks
 [/INST] The problem lies in
$$\frac{1}{(4+9x)^{b+1}}\sim \frac{1}{9x^{b+1}}.$$
It should be
$$\frac{1}{(4+9x)^{b+1}}\sim \frac{1}{(9x)^{b+1}}$$
because
$$\lim_{x\to\infty}\frac{(9x)^{b+1}}{(4+9x)^{b+1}}=1.$$
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