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Compute $\lim \limits_{x \to \infty} \frac{(2x+3)^x-3^x}{2^x-3^x}$ without L’Hospital’s rule

How can I compute $$\lim\limits_{x\to\infty}\frac{(2x+3)^x-3^x}{2^x-3^x}$$ without L’Hospital’s rule?

A:

$(\forall x\ge 1)$ $(2x+3)^x=e^{x\log(2x+3)}$.
We have $-x\log(2x+3)\ge \log(2x+3)-1\ge\log3$.
Thus, $(\forall x\ge 1)$ $(2x+3)^x-3^x\le\exp(\log3-x\log(2x+3))-3^x=(3-2\log3)\exp(\log3-x\log(2x+3))$.
Now, $\lim_{x\to\infty}(3-2\log3)\exp(\log3-x\log(2x+3))=3-2\log3$ and hence $\limsup_{x\to\infty}(2x+3)^x-3^x\le 3-2\log3$.

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