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.