What is the limit of the combination of these two functions If the functions $f$ and $g$ deﬁned on the interval $[0,2]$ satisfy  $ \lim_{x\to 1}$  f(x) = 1 and $\lim_{x\to 1} g(x) = −1$, then, is it true that $\lim_{x\to 1} (2f + g + fg)(x) = 0$ ? 
$ \lim_{x\to 1} 2f(x)=2$, limit of $ \lim_{x\to 1}g (x)=-1$, and $ \lim_{x\to 1}fg (x)=-1$, so I am saying yes this is true. 

Yes, the answer is correct.
$$\lim_{x\rightarrow a}( f(x)+g(x))=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$$
if $\lim_{x\rightarrow a}f(x)$ and $\lim_{x\rightarrow a}g(x)$ exist.
This can be shown as follows:
Let $\lim_{x\rightarrow a}f(x)=L_1$ and $\lim_{x\rightarrow a}g(x)=L_2$
Given $\epsilon\gt0$, there exist $\delta_1,\delta_2\gt0$ such that
$$|f(x)-L_1|\lt\epsilon$$ and $$|g(x)-L_2|\lt\epsilon$$
for all $|x-a|\lt\min(\delta_1,\delta_2)$.
Thus,
$$|f(x)+g(x)-L_1-L_2|\le|f(x)-L_1|+|f(x)-L_2|<\epsilon+\epsilon=2\epsilon$$
for all $|x-a|\lt\min(\delta_1,\delta_2)$.
Hence,
$$\lim_{x\rightarrow a}( f(x)+g(x))=L_1+L_2$$
Note that the converse is not true.
It can be similarly shown that given the individual limits,
$$\lim_{x\rightarrow a}f(x)g(x)=\lim_{x\rightarrow a}f(x)\times\lim_{x\rightarrow a}g(x)$$
And,
$$\lim_{x\rightarrow a}kf(x)=k\lim_{x\rightarrow a}f(x)$$ for a constant $k$.
