Limit existence proof

Question

Suppose I have limit $$ \lim_{n\to \infty }\left(f(n)+g_1(n)\right)=k \tag{1} $$ And $$ \lim_{n\to \infty }\frac{g_1(n)}{g_2(n)}=1 \tag{2} $$

Can I conclude that following limit exists $$\lim_{n\to \infty }\left(f(n)+g_2(n)\right)=k$$

If so how can i prove this?

My attempt

Add (1) to (2) and use limit sum law $$ \lim_{n\to \infty }\left(f(n)+g_1(n)+\frac{g_1(n)}{g_2(n)}\right)=k+1 $$ $$ \lim_{n\to \infty }\left(f(n)+\frac{g_1(n) g_2(n)}{g_2(n)}+\frac{g_1(n)}{g_2(n)}\right)=k+1 $$ $$ \lim_{n\to \infty }\left(f(n)+1 g_2(n)\right)+1=k+1 $$ And finally $$ \lim_{n\to \infty }\left(f(n)+g_2(n)\right)=k $$ Is this correct?

Answer 1

It isn't true. Take $f(n)=n^2$, $g_1(n)=-n^2+k$, and $g_2(n)=-n^2+n$. We have $$\lim_{n\to\infty}f(n)+g_1(n)=k$$ and $$\lim_{n\to\infty}\frac{g_1(n)}{g_2(n)}=1,$$ but $$\lim_{n\to\infty}f(n)+g_2(n)=\infty.$$

Your mistake lies in the second-to-last displayed equation where you take the limit of just one part of the second term... this implicitly assumes convergence of the other sequences when you don't necessarily have that convergence.