The following functions are undefined at x =0. For each determine if this point of discontinuity is removable or not If, so state the value for the function at x = 0 which allows for a continuous extension.(a) f ( x ) = 1 / ( sin ⁡ ( x ) )(b) f ( x ) = x sin ⁡ ( 1 / x )(c) f ( x ) = ( | sin ⁡ ( x ) | ) / x(d) f ( x ) = ( sin ⁡ ( 2 x ) ) / x

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The following functions are undefined at x =0. For each determine if this point of discontinuity is removable or not If, so state the value for the function at x = 0 which allows for a continuous extension.(a)   f  (  x  )  =  1      /    (  sin  ⁡  (  x  )  )(b)   f  (  x  )  =  x  sin  ⁡  (  1      /    x  )(c)   f  (  x  )  =  (      |    sin  ⁡  (  x  )      |    )      /    x(d)  f  (  x  )  =  (  sin  ⁡  (  2  x  )  )      /    x

Desirae Wu · Accepted Answer

a_) We have f(x) =        1          sin      ⁡      x            lim          x      →      0        f  (  x  )  =      lim          x      →      0            1          sin      ⁡      x      , which does not exist. Hence the discontinuity is not able to be removed.b) we have   f  (  x  )  =  x  sin  ⁡      1    x        lim          x      →      0        f  (  x  )  =      lim          x      →      0        x  sin  ⁡      1    x    =  0since the limit exists at x=0 so the function has removable discontinuity at  x=0.By defining f(0)= 0 at x=0 the function becomes contiuous at 0c)we have   f  (  x  )  =                    |            sin      ⁡      x              |              x          lim          x      →              0        −              f  (  x  )  =      lim          x      →              0        −                        sin      ⁡      x        x    =  −  1        lim          x      →              0        +              f  (  x  )  =      lim          x      →              0        +                        sin      ⁡      x        x    =  1The function has different left and right limits at 0, so the discontinuity is not able to be removed.d) We have   f  (  x  )  =            sin      ⁡      2      x        x          lim          x      →      0        f  (  x  )  =      lim          x      →      0                  sin      ⁡      2      x        x    =  2since the limit exists at x=0 so the function has removable discontinuity at x=0. Now by defining f(0)=2 at x=0 the function becomes continuious at 0a)not  removableb)removablec)not removabled) removable

The following functions are undefined at x =0. For each determine if this point of discontinuity is removable or not If, so state the value for the function at x = 0 which allows for a continuous extension. (a) f(x) = 1 /(sin(x)) (b) f(x) = x sin(1/x) (c) f(x) = (|sin(x)|)/x (d)f(x) = (sin(2x))/x

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