Certainly! Here's a graduate-level tricky question on limits and continuity:

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Problem:

Let $f(x)$ be a continuous function such that

$\lim_{x \to a} \frac{f(x) - f(a)}{x - a} = 2$

for some $a \in \mathbb{R}$. Additionally, it is given that $f'(a) = 2$ and that $f$ is differentiable at every point in its domain.

Now, define a new function $g(x)$ as follows:

$g(x) = \frac{f(x) - f(a)}{x - a}$

for $x \neq a$, and $g(a) = \lim_{x \to a} g(x)$ if the limit exists.

Find the value of $g(a)$.

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This problem involves understanding the behavior of the difference quotient and continuity, and applying limits and the concept of differentiability.

Would you like a detailed explanation for how to solve it? Let me know if you have any questions!

Related Questions:

1. How does the concept of the difference quotient relate to the definition of the derivative?
2. What is the connection between the limit of a function and its differentiability at a point?
3. Can a function be continuous but not differentiable? Provide an example.
4. What would happen if the given limit for $f(x)$ at $a$ were not finite?
5. How can we use the continuity of a function to determine its behavior at points of non-differentiability?

Tip: In problems involving limits and derivatives, remember that continuity and differentiability at a point imply specific behaviors of the function around that point. Use this knowledge to simplify your analysis.