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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY1 Mark
Question
The function $\text{f(x)}=\frac{4-\text{x}^2}{4\text{x}-\text{x}^3}$
  1. Discontinuous at only one point.
  2. Discontinuous exactly at two points.
  3. Discontinuous exactly at three points.
  4. None of these.
✓

Answer

  1. Discontinuous exactly at three points.

Solution:

Given,

$\text{f(x)}=\frac{4-\text{x}^2}{4\text{x}-\text{x}^3}$

$\Rightarrow\text{f(x)}=\frac{4-\text{x}^2}{\text{x}(4-\text{x}^2)}$

$\Rightarrow\text{f(x)}=\frac{1}{\text{x}},\text{x}\neq0 $ and  $4-\text{x}^2\neq0 $  or $ \text{x}\neq0,\pm2$

Clearly, f(x) is defined and continuous for all real numbers except $\left\{0,\pm2\right\}$

Therefore, f(x) is discontinuous exactly at three points.

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