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Question

Let fx=1-cos xx2, whenx0 1 , whenx=0
Show that f(x) is discontinuous at x = 0.

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Solution

Given:
fx=1-cosxx2, when x01, when x=0

Consider:
limx0fx=limx01-cosxx2limx0fx=limx02sin2x2x2limx0fx=limx02sin2x24x24limx0fx=limx02sinx224x22limx0fx=24limx0sinx2x22limx0fx=12·12=12

Given:
f0=1

∴ ​limx0fxf0

Thus, f(x) is discontinuous at x = 0
.

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