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One-sided limits in calculus are crucial for analyzing how functions behave as they approach a specific point from one direction. This concept helps determine the existence of limits, continuity of functions, and the presence of vertical asymptotes. For example, a piecewise function with different values on either side of a point shows how one-sided limits can differ, indicating a discontinuity. Similarly, functions that grow unbounded near a point reveal vertical asymptotes through one-sided limits.

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## Definition of One-Sided Limits

### Left-Sided Limit

A left-sided limit examines the values of a function as x approaches a point from the left side, but does not include the point itself

### Right-Sided Limit

A right-sided limit considers the values of a function as x approaches a point from the right side

### Existence of Limits

For a limit to exist at a point, both the left-sided and right-sided limits must converge to the same value

## Evaluating One-Sided Limits

### Using Graphs

One-sided limits can be evaluated by observing the trend of a function on a graph as it approaches a point from one side

### Using Tables of Values

One-sided limits can also be evaluated by examining the numerical values of a function as it approaches a point from one side

### Real-World Examples

One-sided limits have practical significance in analyzing the behavior of functions near specific points

## Applications of One-Sided Limits

### Continuity of Functions

One-sided limits are crucial for understanding the continuity of functions and their behavior at points where they may not be defined or where their behavior changes abruptly

### Vertical Asymptotes

One-sided limits are instrumental in identifying vertical asymptotes, which are characterized by a function's values increasing without bound as it approaches a certain point

### Discontinuity

One-sided limits can also help identify points of discontinuity in functions

Algorino

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