Question

Existence of the limit:
Answer

Answer 1

a) From the left, the limit is 1, since

`\displaystyle\lim_(x\to1^-)f(x)=\lim_(x\to1)x=1`

b) From the right, it's 2, since

`\displaystyle\lim_(x\to1^+)f(x)=\lim_(x\to1)2=2`

c) Also 2, same reasoning as in (b).

d) 1, since

`\displaystyle\lim_(x\to4^+)f(x)=\lim_(x\to4)(5-x)=1`

e) No, neither two-sided limits exist because the respective one-sided limits do not agree.