AP Calculus AB Exam Question:

Let f(x) be a continuous function defined on the closed interval [a, b], where a < b. Determine the limit of the function as x approaches c, where c is a real number within the interval (a, b), using the $\epsilon$-$\delta$ definition of a limit.

Consider the function:

Using the $\epsilon$-$\delta$ definition of a limit, determine the following limit:

Step-by-Step Solution:

To find the limit as x approaches c, we will employ the $\epsilon$-$\delta$ definition of a limit, which states:

For a given value $\epsilon > 0$, there exists a value $\delta > 0$ such that if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$, where L is the desired limit.

First, let's analyze the behavior of the function on the left and right sides of c.

For x values less than or equal to c:

For x values greater than c:

Now, let's find the limit of f(x) as x approaches c from the left side.

Similarly, let's find the limit of f(x) as x approaches c from the right side.

Next, let's combine both limits:

Therefore, the limit of the given function as x approaches c is $\lim_{x \to c} f(x) = c^2$.