Given,
Function is f(x) = \(\begin{cases} x^3 +3, \quad \text{if} \,x\ne 0 \\ 1, \,\,\,\,\,\,\,\,\,\,\,\,\,\quad \text {if}\,x=0 \end{cases}\)
If f(x) is continuous at x = 0 then
\(\lim\limits_{x \to 0^ -} f(x) = \lim\limits_{x \to 0^ +}\,f(x) =f(0)\)
Finding L.H.L.
\(\lim\limits_{x \to 0^ -}\,x^ 3 +3\)
\(=\lim\limits_{h \to 0} (0-h)^ 3 + 3\)
\(=\lim\limits_{h \to 0} (-h)^ 3+3\)
Putting h = 0 then we get,
= (−0)3 + 3 = 0 + 3 = 3
Finding R.H.L.
\(\lim\limits_{x \to 0^+} x^3 + 3\)
\(=\lim\limits_{h \to 0} \,(0+h)^ 3 + 3\)
\(=\lim\limits_{h \to 0}\,(h)^3 + 3\)
Putting h = 0 then we get,
= (0)3 + 3 = 0 + 3 = 3
To find f(x) at x=0
f(x) = 1 at x = 0
⇒ f(0) = 1
Hence,
\(\lim\limits_{x \to 0^ -} f(x) = \lim\limits_{x \to 0^ +}\,f(x) \ne(0)\)
Therefore, the function f(x) = \(\begin{cases} x^3 +3, \quad \text{if} \,x\ne 0 \\ 1, \,\,\,\,\,\,\,\,\,\,\,\,\,\quad \text {if}\,x=0 \end{cases}\) is not continuous at x = 0
