Q101 / 160 · AP EAMCET 2024 (19 May Shift 2) · Mathematics

The function $f(x)=\left\{\begin{array}{ll}\frac{2}{5-x}, & x \lt 3 \\ 5-x, & x \geq 3\end{array}\right.$ is

  • Correct answer
    A
    left discontinuous at $x=3$
  • B
    left continuous at $x=3$
  • C
    right discontinuous at $x=5$
  • D
    discontinuous at $x=5$

Solution

Given, $f(x)= \begin{cases}\frac{2}{5-x}, & x \lt 3 \\ 5-x, & x \geq 3\end{cases}$ $\begin{aligned} & \because \lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{-}} \frac{2}{5-x}=\frac{2}{5-3}=1 \\ & \\ & \lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3^{+}}(5-x)=2 \text { and } f(3)=5-3=2 \end{aligned}$ Since, $\lim _{x \rightarrow 3^{-}} f(x) \neq \lim _{x \rightarrow 3^{+}} f(x)=f(3)$ So, $f(x)$ is left discontinuous at $x=3$.

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

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