f x - left matrix-frac -x-4-2x-4- - if - x n e q 4 l0-8 if - X -4Is the function f x continuous at x -0 -

We have, f(x)={|x 4|/2(x 4), amp; if x≠ 4 0, amp; if x=4 . At x=4, LHL= x→ 4^ lim|x 4|/2(x 4) =h→ 0lim|4 h 4|/2(4 h) 4=h→ 0lim|0 h|/(8 2h 8) =h→ ...

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Standard XII

Mathematics

Definition of Functions

f x = left ma...

Question

$f (x) = ⎧ ⎨ ⎩ \begin{matrix} \frac{| x - 4 |}{2 (x - 4)}, & i f x \neq 4 0, & i f x = 4 \end{matrix}$

Is the function f(x) continuous at x=0?

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Solution

We have, $f (x) = ⎧ ⎨ ⎩ \begin{matrix} \frac{| x - 4 |}{2 (x - 4)}, & i f x \neq 4 0, & i f x = 4 \end{matrix}$

At x=4, $L H L = l i m x \to 4^{-} \frac{| x - 4 |}{2 (x - 4)} = l i m h \to 0 \frac{| 4 - h - 4 |}{2 (4 - h) - 4} = l i m h \to 0 \frac{| 0 - h |}{(8 - 2 h - 8)} = l i m h \to 0 \frac{h}{- 2 h} = \frac{- 1}{2} f (4) = 0 \neq L H L$

so, f(x) is discontinuous at x=4

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f(x)=⎧⎨⎩|x−4|2(x−4),if x≠40,if x=4 Is the function f(x) continuous at x=0?

We have, f(x)=⎧⎨⎩|x−4|2(x−4),if x≠40,if x=4 At x=4, LHL=limx→4−|x−4|2(x−4)=limh→0|4−h−4|2(4−h)−4=limh→0|0−h|(8−2h−8)=limh→0h−2h=−12f(4)=0≠LHL so, f(x) is discontinuous at x=4

$f (x) = ⎧ ⎨ ⎩ \begin{matrix} \frac{| x - 4 |}{2 (x - 4)}, & i f x \neq 4 0, & i f x = 4 \end{matrix}$

Is the function f(x) continuous at x=0?