If a function is defined in pieces, and if the definition changes
at x=a, then we use the definition for x<a to compute
limx→a−f(x), we use the definition at
x=a to compute f(a), and the definition for x>a to
compute limx→a+f(x), and then we
compare the three quantities.
Example: If f(x)={1−xx<0,x2x≥0, then
Then limx→0−f(x)=limx→0−(1−x)=1,limx→0+f(x)=limx→0+(x2)=0, and f(0)=02=0.
DO: Check that the
values above are correct, using the given piecewise definition
of f.
Since the limits from the left and right do not agree, the limit
does not exist, and the function is discontinuous at x=0.
DO:Consider the
limits above, and try to sketch what happens to the graph of fnear
x=0. Don't worry about other parts of the function;
use only the limits as x→0− and as x→0+, and f(0),
not the definition of the function.