Maharashtra State BoardHSC Science (Electronics) 11th
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Examine the continuity of f(x) =sinx, for x≤π4=cosx, for x>π4} at x=π4 - Mathematics and Statistics

Sum

Examine the continuity of `"f"(x)  {:(= sin x",",  "for"  x ≤ pi/4), (= cos x",",  "for"  x > pi/4):}}  "at"  x = pi/4`

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Solution

`"f"(x)  {:(= sin x";",   x ≤ pi/4), (= cos x";",  x > pi/4):}` 

`lim_(x -> pi^-/4) "f"(x) = lim_(x -> pi^-/4) (sin x)`

= `sin  pi/4`

= `1/sqrt(2)`

`lim_(x -> pi^+/4) "f"(x) = lim_(x -> pi^+/4) (cos x)`

= `cos  pi/4`

= `1/sqrt(2)`

Also `"f"(pi/4) = sin  pi/4`

= `1/sqrt(2)`

∴ `lim_(x -> pi^-/4) "f"(x) = lim_(x -> pi^+/4) "f"(x) = "f"(pi/4)`

∴ f(x) is continuous at x = `pi/4`

  Is there an error in this question or solution?
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