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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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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
A right-sided limit considers the values of a function as x approaches a point from the right side
For a limit to exist at a point, both the left-sided and right-sided limits must converge to the same value
One-sided limits can be evaluated by observing the trend of a function on a graph as it approaches a point from one side
One-sided limits can also be evaluated by examining the numerical values of a function as it approaches a point from one side
One-sided limits have practical significance in analyzing the behavior of functions near specific points
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
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
One-sided limits can also help identify points of discontinuity in functions
00
For a function's limit to be defined at a point, the ______ and ______ limits must converge to the same value.
left-sided
right-sided
01
Definition of one-sided limits
Left-sided limit: value that function approaches as x nears a from the left. Right-sided limit: value function approaches as x nears a from the right.
02
Role of one-sided limits in continuity
A function is continuous at point a if one-sided limits at a are equal and equal to the function's value at a.
