Q100 / 160 · AP EAMCET 2023 (19 May Shift 1) · Mathematics

If a function $f(x)$ defined on $[a, b]$ is discontinuous at $x=\alpha \in(a, b)$, then

  • A
    $\lim _{x \rightarrow \alpha^{-}} f(x)=\lim _{x \rightarrow \alpha^{+}} f(x)=f(\alpha)$
  • Correct answer
    B
    $\lim _{x \rightarrow \alpha^{+}} f(x) \neq f(\alpha)$
  • C
    $\lim _{x \rightarrow a^{-}} f(x)=f(a)$
  • D
    $\lim _{x \rightarrow b^{+}} f(x)=f(b)$

Solution

$f(x)$ is defined on $[a, b]$ and discontinuous at $x=\alpha \in(a, b)$ Since $f(x)$ is discontinuous at $x=\alpha \in(a, b)$ Hence option (a) will never be true but $\lim _{x \rightarrow \alpha} f(x) \neq f(\alpha)$ is showing that $f(x)$ is discontinuous at $x=\alpha \in(a, b)$ Hence option (b) is correct. Since function is not defined for $x \rightarrow a^{-}$. Hence option (c) is incorrect. Since function is not defined when $x \rightarrow b^{+}$. Hence (d) is incorrect.

Hence, the correct answer is option: 'B'.

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