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Continuity is a fundamental concept in calculus that describes a function without any abrupt changes, gaps, or jumps. A function is continuous at a point if the following three conditions are met:

• The function is defined at that point.
• The limit of the function exists as the point is approached.
• The value of the function at that point equals the limit.

When these criteria are met, we can say the function is continuous at that point, ensuring a smooth and uninterrupted graph. In our earlier example with the function \( f(x) = \frac{x}{|x|} \), we have determined that the function is not continuous at \( x = 0 \). This conclusion is drawn because the left-hand limit and right-hand limit at this point are not equal. Hence, the function doesn't fulfill all the necessary criteria for continuity at \( x = 0 \).
Understanding continuity helps in analyzing and predicting the behavior of functions across different points.

Absolute value, represented by \( |x| \), is a crucial mathematical concept used to determine the non-negative magnitude of a real number, regardless of its sign. Essentially, absolute value takes a negative number and outputs its positive counterpart. For example:

• If \(x = 3\), then \(|x| = 3\).
• If \(x = -3\), then \(|x| = 3\).

The absolute value plays a significant role in the function \( f(x) = \frac{x}{|x|} \). Here, the absolute value ensures the denominator is positive, whether \(x\) is positive or negative. This creates different expressions for the function based on the value of \(x\):

• For \(x > 0\), \(|x| = x\), making the function \( f(x) = \frac{x}{x} = 1\).
• For \(x < 0\), \(|x| = -x\), so the function becomes \( f(x) = \frac{x}{-x} = -1\).

Understanding absolute value allows us to analyze functions thoroughly, especially when evaluating one-sided limits.

Left-hand limit refers to the limit of a function as the variable approaches a specific value from the left, or from values less than the point in question. When examining \( \lim_{x \to 0^{-}} f(x) \), we observe the behavior of the function as \(x\) moves towards zero from negative values. To illustrate:For \( f(x) = \frac{x}{|x|} \), when approaching from the left:

• \( x < 0 \), thus \(|x| = -x\).
• The function becomes \( f(x) = \frac{x}{-x} = -1 \).

This means as \(x\) approaches zero from the negative side, \( f(x) \) consistently outputs \(-1\). Thus, \( \lim_{x \to 0^{-}} f(x) = -1 \).
Understanding one-sided limits helps determine the behavior of a function on a particular side of a point, which is essential for analyzing limits and continuity.

Right-hand limit explores the behavior of a function as the variable approaches a designated value from the right, meaning from values greater than the point in question. In scenarios like \( \lim_{x \to 0^{+}} f(x) \), we look at how the function performs as \(x\) nears zero from positive values. In the function \( f(x) = \frac{x}{|x|} \):

• For \( x > 0 \), \(|x| = x\).
• The function transforms to \( f(x) = \frac{x}{x} = 1 \).

Thus, for \(x\) approaching zero from the positive side, \( f(x) \) yields \(1\) consistently. Therefore, \( \lim_{x \to 0^{+}} f(x) = 1 \).
Grasping the concept of right-hand limits allows students to predict a function’s behavior approaching a specific value from the right side, aiding in the holistic understanding of the function's properties.