1. limits

http://mustafayagci.com.tr/PDF/MY_M3.pdf

Page 152. Last question on the right side at the bottom of the page. Page 152 last question at the right side of the page.The last question in the bottom right.
I couldn't understand the operation in the answer.

2. Re: limits

I do not understand your question. I do not see any question where "he adds 2 to the limit sign". On page 152, the last question on the right side at the bottom of the page appears to be:

$\displaystyle \lim_{x \to 1^-} \dfrac{f(x+2)}{f(2-x)}$

And there is a graph of $f(x)$.

Perhaps you are referring how he goes from
$\displaystyle \lim_{x \to 1^-} \dfrac{f(x+2)}{f(2-x)} = \dfrac{\displaystyle \lim_{x \to 3^-} f(x)}{\displaystyle \lim_{x \to 1^+} f(x)}$

That is because as $x$ approaches 1 from the left, $x+2$ approaches 3 from the left. So, he reevaluates the limit of the numerator as $\displaystyle \lim_{x \to 3^-} f(x)$. As $x$ approaches 1 from the left, $2-x$ approaches 1 from the right. So, he reevaluates the limit of the denominator as $\displaystyle \lim_{x \to 1^+} f(x)$.

Many Thanks.