How do you find the limit as x approaches 2 of the piecewise function #f(x)={(4-x^2,if x<=2),(x-1,if x>2):}# ?

1 Answer
Wataru
Sep 7, 2014

Since the left-hand limit and the right-hand limit at #x=2# do not match, the limit #lim_{x to 2}f(x)# does not exist.

Let us find the left-hand limit.
#lim_{x to 2^-}f(x)=lim_{x to 2^-}(4-x^2)=4-(2)^2=0#

Let us find the right-hand limit.
#lim_{x to 2^+}f(x)=lim_{x to 2^+}(x-1)=(2)-1=1#

Since the above one-sided limits at #x=2# do not match, the limit at #x=2# does not exist.

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