I want to know if I can "use" a limit after I've used L'hopital's rule on it? I'm not sure how to better word it, but I can show you what I tried, maybe you can tell me if it is right or why it is wrong.
$$\lim_{x \to \infty} e^x – \frac{e^x}{x+1}$$
We can split this into two limits
$$\lim_{x \to \infty} e^x – \lim_{x \to \infty} \frac{e^x}{x+1}$$
Now since the limit on the right side is infinity over infinity, we can apply L'Hopital's rule
$$\lim_{x \to \infty} e^x – \lim_{x \to \infty} \frac{e^x}{1}$$
Now we can join the two limits back (I am "reusing" the limit after applying L'hopital…is this allowed?)
$$\lim_{x \to \infty} e^x – e^x$$
Subtracting we have
$$\lim_{x \to \infty} 0 = 0$$
Best Answer
The theorem is $\lim_{x \to a} \ (f(x) + g(x)) = \lim_{x \to a} \ f(x) + \lim_{x \to a} \ g(x)$ is valid in general, if both limits $\lim_{x \to a}\ f(x) \ \text{and} \lim_{x \to a}\ g(x)$ are individually finitely exists.