Consider: $$\lim_{x \to \infty} \left(x - \ln(e^x + e^{-x})\right)$$
I wasn't sure how to treat the $\infty - \infty$ property. Can I exponentiate the function to get $$e^x - (e^x + e^{-x}) = \frac{1}{e^x}$$ $$\lim_{x \to \infty} \frac{1}{e^x} = 0$$
I feel like I have ignored the limit part of the expression when exponentiating. Do I have to exponentiate the limit expression as well when I do this or can I ignore it for the moment?