Hello Dear,

Well, to check for limit, its left hand limit, right hand limit and actual limit must all be equal, So,

for x=1, f(x)=3-x; since f(x)=3-x for x<2 (and obviously 1<2)

So, for Right Hand Limit f(1+h)=2-h=2 (putting h=0, where h is very small and positive)

Similarly for Left hand Limit f(1-h)=2+h = 2 (and putting h=0, where h is very small and positive)

And at x=1, f(1)=2;

Hence, all three limits exist and are equal, so lim x--> 1 for f(x) exists.

(My opinion, its limit is changing at x=2, either you have miswritten this question, or this question is way easier and trying to say that limit doesn't get affected if function doesn't change)

Hope this helps, and feel free to ask any further query...