Question

which of the following statements, if true, can be used to conclude that ( f(2) ) exists?
i. ( lim_{x
ightarrow2}f(x) ) exists.
ii. ( f ) is continuous at ( x = 2 ).
iii. ( f ) is differentiable at ( x = 2 ).
a i only
b ii only
c ii and iii only
d i, ii, and iii

Explanation:

• Statement I: The existence of \(\lim_{x

ightarrow2}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

ightarrow 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