Doubt about asymptotic analysis $\frac{1}{x^{a}*(4+9x)^{b+1}}\sim \frac{1}{9^{b+1}x^{a+b+1}}$

Question

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

Answer 1

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 $$\lim_{x\to\infty}\frac{(9x)^{b+1}}{(4+9x)^{b+1}}=1.$$