Hello, realintegerz!
The first problem is straightforward.
Exactly where is your difficulty?
Let: .$\displaystyle f(x) \:=\:\begin{Bmatrix} x & x < \text{}1 \\
x 2 & \text{}1 \le x < 1 \\
1 & x = 1 \\
(x1)^2 & x>1 \end{Bmatrix}$
$\displaystyle a)\;\;\lim_{x\to\,\text{}1^\text{}}f(x) $ $\displaystyle \lim_{x\to\,\text{}1^}f(x) \;=\;\lim_{x\to\,\text{}1^}(x) \;=\;(\text{}1) \;=\;1$
$\displaystyle b)\;\;\lim_{x\to\text{}1^+}f(x)$
$\displaystyle \lim_{x\to\,\text{}1^+}f(x) \;=\;\lim_{x\to\,\text{}1^+}(x2) \;=\;\text{}1  2 \:=\:3$
$\displaystyle c)\;\;\lim_{x\to0}f(x)$
$\displaystyle \lim_{x\to0}f(x) \;=\;\lim_{x\to0}(x2) \;=\;02 \;=\;2$
$\displaystyle \lim_{x\to1^}f(x)$
$\displaystyle \lim_{x\to1^}f(x) \;=\;\lim_{x\to1^}(x2) \;=\;12 \;=\;1$
$\displaystyle e)\;\;\lim_{x\to1^+}f(x)$
$\displaystyle \lim_{x\to1^+}f(x) \;=\;\lim_{x\to1^+}(x1)^2 \;=\;(11)^2 \;=\;0$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
A sketch makes the limits quite clear . . . Code:
* 
*  *
* 
*  *
o  * *
 *
       +  o     

 o
 *
*
* 
* 