In order to calculate the limit of the function for a specific value of x, you just need to apply this value of x in the equation.
If the limit says "x -> 1-", it means we are approaching the value of 1 by the left (that is, from 0 to 1 for example).
If the limit says "x -> 1+", it means we are approaching the value of 1 by the right (that is, from 2 to 1 for example).
a)
To calculate this limit, we use the first part of the piecewise function, as we are approaching by the left (values lesser than 1):
[tex]\begin{gathered} g(x)=2-x \\ \lim _{x\to1^-}g(x)=2-1=1 \end{gathered}[/tex]b) To calculate this limit, we use the second part of the piecewise function, as we are approaching by the right (values greater than 1):
[tex]\begin{gathered} g(x)=\frac{x}{2}+1 \\ \lim _{x\to1^+}g(x)=\frac{1}{2}+1=\frac{3}{2} \end{gathered}[/tex]c) As the limit at 1 from the left is diferent from the limit at 1 from the right, the limit at 1 does not exist (it's undefined).
d) For x = 1, we need to use the first part of the piecewise function (because x = 1 is inside the inverval x <= 1), so we have:
[tex]\begin{gathered} g(x)=2-x \\ g(1)=2-1=1 \end{gathered}[/tex]