Limits of picewise functions

Question

i have two piecewise functions

f(x) = 1 if x < 1

          x if 1< x

 

g(x) = x if not equals 2

          3 if x = 2

 

lim_{x approaches 2}{ f(x) + g(x)}

my answer here is that the limit if f(x) 

doesnt exist because approaching the limit 2 from the left and the right gives me different results. approaching from the left i get 1 , approaching from the right i get 2.

is this correct? 

And how would i solve the g(x) limit?

 

Answer 1

CONCERNING limit of g(x) at 2: this limit is 2. The fact that g(2) = 3 has no effect on the limit.  (In fact, if if the limit at 2 and f(2) are equal, f would be continuous.)

 

Therefore limit as x approaches 2{ f(x) + g(x)} = 2 + 2