[Math] Am I allowed to apply L’Hospital’s Rule inside of the natural logarithm function

Question

I have the following limit:

$$\lim_{x\rightarrow \infty} \ln\left(\frac{2x^2+1}{x^2+1}\right)$$

If I was finding the limit of only the terms inside the natural log function, I would have the indeterminate form: $$\frac{\infty}{\infty}$$

I want to know if I am allowed to apply L'Hospital's Rule to only the inside terms while ignoring the natural logarithm function, giving me the answer:$$\ln\left(\frac{4x}{2x}\right)=\ln(2)$$

Answer 1

For a continuous function $f$ on $a\in\overline{\Bbb R}$ we have $$\lim_{x\to a}f(g(x))=f(\lim_{x\to a}g(x))$$