The correct answer is:

A) $\lim_{x \to 1} f(x)$ exists.

Explanation:

The existence of a limit at $x = 1$ does not guarantee that the function $f$ is defined at $x = 1$. For a function to be defined at a point, $f(1)$ must exist. However, the limit $\lim_{x \to 1} f(x)$ only describes the behavior of the function as $x$ approaches 1 and does not tell us whether $f(1)$ is actually defined.

The other options:

• B) $f$ is continuous at $x = 1$: Continuity implies that $f$ is defined at $x = 1$ and $\lim_{x \to 1} f(x) = f(1)$.
• C) $f$ is differentiable at $x = 1$: Differentiability implies continuity, which means $f$ must be defined at $x = 1$.
• D) The tangent line exists at $x = 1$: For a tangent line to exist, the function must be differentiable, which also means $f$ is defined at $x = 1$.

Thus, A is the only statement that cannot be used to conclude that $f$ is defined at $x = 1$.

Would you like a further breakdown of limits and continuity? Here are related questions to consider:

1. What are the formal definitions of limits and continuity?
2. How does differentiability imply continuity?
3. Can a function have a limit at $x = 1$ but not be continuous there?
4. How do you check if a tangent line exists for a function?
5. What is an example of a function where the limit exists, but the function is undefined at that point?

Tip: Remember that the existence of a limit does not mean the function value at that point exists! Always check if $f(a)$ is defined explicitly.