Show that the function f given by f(x)= ll x^ 3 +3, & if x 0 1, &… - Vidyadip

Question

Show that the function f given by $f(x)=\left\{\begin{array}{ll} {x^{3}+3,} & {\text { if } x \neq 0} \\ {1,} & {\text { if } x=0} \end{array}\right.$ is not continuous at x = 0.

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Answer

The function is defined at x = 0 and f(0) =1.
When x $\neq$ 0, the function is given by a polynomial. Hence,
$\mathop {\lim }\limits_{x \to 0} f(x) = \mathop {\lim }\limits_{x \to 0} $ (x³ + 3) = 0³ + 3 = 3
Since , the limit of f at x = 0 does not coincide with f(0), the function is not continuous at x = 0. It may be noted that
x = 0 is the only point of discontinuity for this function.

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