3) 8pt sketch a graph of a function ( y = f(x) ), where ( f ) is continuous for all real numbers except 2…

3) 8pt sketch a graph of a function ( y = f(x) ), where ( f ) is continuous for all real numbers except 2 and -4, and ( f ) satisfies the following conditions:\n( lim _{x \rightarrow 2^{-}} f(x)=infty )( lim _{x \rightarrow 2^{+}} f(x)=3 )( lim _{x \rightarrow-4} f(x)=-2 )( f(-4)=1 )( f(2)=3 )( f(0)=0 )\nlabel the axes and indicate the scale on the axes.\nclassify the types of discontinuities at 2 and at -4 (jump, removable, or infinite).\n( f ) has a ________ discontinuity at ( x = -4 ).\n( f ) has a ________ discontinuity at ( x = 2 ).

3) 8pt sketch a graph of a function ( y = f(x) ), where ( f ) is continuous for all real numbers except 2 and -4, and ( f ) satisfies the following conditions:\n( lim _{x \rightarrow 2^{-}} f(x)=infty )( lim _{x \rightarrow 2^{+}} f(x)=3 )( lim _{x \rightarrow-4} f(x)=-2 )( f(-4)=1 )( f(2)=3 )( f(0)=0 )\nlabel the axes and indicate the scale on the axes.\nclassify the types of discontinuities at 2 and at -4 (jump, removable, or infinite).\n( f ) has a ________ discontinuity at ( x = -4 ).\n( f ) has a ________ discontinuity at ( x = 2 ).

Answer

Brief Explanations:

  • For (x = - 4):
    • We know that (\lim_{x\rightarrow - 4}f(x)=-2) and (f(-4) = 1). Since (\lim_{x\rightarrow - 4}f(x)) exists (a finite value) but (\lim_{x\rightarrow - 4}f(x)\neq f(-4)), by the definition of a removable discontinuity (a discontinuity where the limit of the function exists at the point but the function value at that point is different from the limit), this is a removable discontinuity.
  • For (x = 2):
    • We have (\lim_{x\rightarrow2^{-}}f(x)=\infty). A discontinuity where the one - sided limit is (\pm\infty) is called an infinite discontinuity.

Answer:

  • (f) has a removable discontinuity at (x=-4).
  • (f) has an infinite discontinuity at (x = 2).