91Ó°ÊÓ

The concept of the left-hand limit is essential in understanding how a function behaves as it approaches a particular point from the left side. In mathematical terms, we denote this as \( \lim_{x \to c^-} f(x) \), meaning that \( x \) approaches the point \( c \) from values less than \( c \).

To calculate the left-hand limit, we use the piece of the function defined for \( x < c \). For instance, in our given exercise involving a piecewise function, the function is \( f(x) = x + 1 \) for \( x < 1 \). As \( x \) gets closer to 1 from the left, \( f(x) \) approaches \( 1 + 1 = 2 \).

Thus, the left-hand limit for this function as \( x \to 1^- \) is 2. This demonstrates how we evaluate the behavior of piecewise functions using the specific definition applicable to the domain of interest.

The right-hand limit investigates the function's behavior as \( x \) approaches a certain point from the right, i.e., from values greater than the point. This is expressed as \( \lim_{x \to c^+} f(x) \).

In our piecewise function example, the segment applicable for \( x > 1 \) is \( f(x) = x - 1 \). As \( x \) gets nearer to 1 from the right, \( f(x) \) tends to \( 1 - 1 = 0 \).

Consequently, the right-hand limit of the function as \( x \to 1^+ \) is 0. By analyzing the right-hand limit, you can determine how the function progresses from the immediate right of the specified point.

Continuity at a point is a fundamental concept ensuring that a function behaves predictably at that point without any discontinuities. A function is continuous at a point \( c \) if the following three conditions are satisfied:

• The function \( f(x) \) is defined at \( x = c \).
• The left-hand limit \( \lim_{x \to c^-} f(x) \) and right-hand limit \( \lim_{x \to c^+} f(x) \) both exist.
• The left-hand and right-hand limits are equal, and both are equal to \( f(c) \).

In our exercise, continuing the function at \( x = 1 \) would mean that \( f(1) = 1 \), yet the two-sided limit \( \lim_{x \to 1} f(x) \) does not exist since \( \lim_{x \to 1^-} f(x) = 2 \) and \( \lim_{x \to 1^+} f(x) = 0 \) are not equal. This absence of equality implies a discontinuity at \( x = 1 \) for this function.

Understanding these conditions will help you recognize points of discontinuity effectively in various mathematical contexts.