Question

which of the following statements, if true, can be used to conclude that ( f(3) ) exists?
i. ( lim_{x \to 3} f(x) ) exists.
ii. ( f ) is continuous at ( x = 3 ).
iii. ( f ) is differentiable at ( x = 3 ).
options:
a) i only
b) ii only
c) ii and iii only
d) i, ii, and iii

Explanation:

• Statement I: The existence of $\lim_{x \to 3} f(x)$ does not guarantee that $f(3)$ exists. A function can have a limit at a point but be undefined there (e.g., a removable discontinuity: $f(x)=\frac{x^2 - 9}{x - 3}$ has $\lim_{x \to 3} f(x)=6$ but $f(3)$ is undefined initially).
• Statement II: A function $f$ is continuous at $x = 3$ if three conditions hold: (1) $f(3)$ exists, (2) $\lim_{x \to 3} f(x)$ exists, and (3) $\lim_{x \to 3} f(x)=f(3)$. So if $f$ is continuous at $x = 3$, $f(3)$ must exist.
• Statement III: Differentiability at a point implies continuity at that point (a theorem in calculus: if $f$ is differentiable at $x = a$, then $f$ is continuous at $x = a$). Since continuity at $x = 3$ requires $f(3)$ to exist, differentiability at $x = 3$ also implies $f(3)$ exists.

Answer:

C. II and III only