which of the following statements, if true, can be used to conclude that ( f(2) ) exists?\ni. (…

which of the following statements, if true, can be used to conclude that ( f(2) ) exists?\ni. ( lim_{x\rightarrow2}f(x) ) exists.\nii. ( f ) is continuous at ( x = 2 ).\niii. ( f ) is differentiable at ( x = 2 ).\na i only\nb ii only\nc ii and iii only\nd i, ii, and iii

which of the following statements, if true, can be used to conclude that ( f(2) ) exists?\ni. ( lim_{x\rightarrow2}f(x) ) exists.\nii. ( f ) is continuous at ( x = 2 ).\niii. ( f ) is differentiable at ( x = 2 ).\na i only\nb ii only\nc ii and iii only\nd i, ii, and iii

Answer

Brief Explanations:

  • Statement I: The existence of (\lim_{x\rightarrow2}f(x)) only tells us about the behavior of the function as (x) approaches (2), not about the value of (f(2)) itself. For example, consider a function with a hole at (x = 2). The limit as (x) approaches (2) can exist, but (f(2)) may not be defined.
  • Statement II: By the definition of continuity, if (f(x)) is continuous at (x=a) (in this case (a = 2)), then (\lim_{x\rightarrow a}f(x)=f(a)). For (f(x)) to be continuous at (x = 2), (f(2)) must be defined.
  • Statement III: If (f(x)) is differentiable at (x=a) (here (a=2)), then (f(x)) is continuous at (x=a). Since differentiability at (x = 2) implies continuity at (x = 2), and continuity at (x = 2) implies (f(2)) exists.

Answer:

C. II and III only