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When discussing the concept of a left-hand limit, we refer to the behavior of a function as it approaches a particular point from the left. In this case, the point of interest is 3. For the function \( F(x) = |x-3| \) when \( x eq 3 \), we examine what happens as \( x \) gets closer to 3 from values less than 3.

To find the left-hand limit, consider that if \( x < 3 \), then \( x-3 \) is negative. The absolute value, \( |x-3| \), will negate this negative value to ensure a positive output. Thus, it behaves as \( |x-3| = -(x-3) \).

As x approaches 3 from the left, \( (x-3) \) moves towards 0. Thus, the left-hand limit is:

• \( \lim_{x \to 3^-} |x-3| = 0 \)

The left-hand limit is essential when determining a function's continuity as it tells us how the function behaves just before reaching the point.

The right-hand limit explores the behavior of the function as it approaches a particular value from the right. For our function, we observe \( F(x) = |x-3| \) when \( x eq 3 \), especially as \( x \) gets close to 3 from values greater than 3.

In this scenario, since \( x > 3 \), \( x-3 \) is positive. The absolute value does not change this, so \( |x-3| = x-3 \). When x nears 3 from the right, this simply approaches 0:

• \( \lim_{x \to 3^+} |x-3| = 0 \)

Examining the right-hand limit is crucial as it helps us understand the function's behavior immediately following the point in question. This complements the left-hand limit to determine overall limit existence.

To determine continuity at a point, the limit definition states that the function's value at that point must equal the limit from both sides. In mathematical terms, a function \( F(x) \) is continuous at a point \( x=c \) if:

• \( \lim_{x \to c^-} F(x) = \lim_{x \to c^+} F(x) = F(c) \)

For this exercise, having the left-hand and right-hand limits at 3 both equal to 0 indicates that a limit exists at \( x=3 \). However, continuity requires that \( F(3) = 0 \), but \( F(3) \) is given as 2.
Because the value of the function at \( x=3 \) (which is 2) does not match the limit (which is 0), \( F(x) \) is not continuous at \( x=3 \). This discrepancy is vital for understanding when functions behave smoothly or have interruptions.

A piecewise function is a function defined by different expressions based on the input's range. In this context, the function \( F(x) \) changes its formula depending on whether \( x=3 \) or not.

• When \( x eq 3 \), \( F(x) = |x-3| \)
• When \( x = 3 \), \( F(x) = 2 \)

Piecewise functions can often have complicated continuity behavior. Each piece can exhibit different properties, and their points of intersection might or might not be continuous. Understanding these types of functions is crucial in identifying discontinuities, as is the case at \( x=3 \) where the function shifts and does not align with the anticipated value from the limit. This study of piecewise functions helps clarify behaviors that aren't obvious from a single uniform equation.