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Evaluate lim sin(x)/x as x approaches 0 - a foundational calculus limit
Key values: sin(x)/x · x -> 0 · Result: 1
Explore the limit of (1 + 1/x)^x as x approaches infinity (Euler's number)
Key values: (1+1/x)^x · x -> Infinity · Result: e
Evaluate the derivative definition limit for x² at x = 3
Key values: (x² - 9)/(x - 3) · x -> 3 · Result: 6
Enter a function f(x), an approach point, and a direction. Click Calculate to see the limit value, approach table, and classification.
A one-sided limit considers the behavior of as approaches a value from only one direction.
Considers only values of less than (approaching from the left).
Considers only values of greater than (approaching from the right).
Key theorem: The two-sided limit exists if and only if both one-sided limits exist and are equal: .
When , the function has a jump discontinuity at . The two-sided limit does not exist, but both one-sided limits are well-defined. Classic examples:
For :
The one-sided limits differ, so the two-sided limit does not exist.
For :
Both one-sided limits equal 1, so the two-sided limit exists and equals 1. Since as well, the function is continuous at .
A one-sided limit examines the behavior of as approaches a value from only one direction. The left-hand limit () is written with a superscript minus, and the right-hand limit () with a superscript plus.
The two-sided limit exists if and only if both one-sided limits exist and are equal. If the left-hand limit and right-hand limit give different values, the two-sided limit does not exist.
A jump discontinuity occurs at a point where the left-hand limit and right-hand limit both exist but are not equal. The function “jumps” from one value to another. Examples include step functions, the floor function at integers, and the sign function at zero.
Use the piece of the function that applies to the direction you are approaching from. For the left-hand limit, use the formula valid for values less than the point. For the right-hand limit, use the formula valid for values greater than the point.
No. A function is continuous at a point only if the two-sided limit exists (meaning both one-sided limits are equal) and equals the function value at that point. If one-sided limits differ, the function has a jump discontinuity and is not continuous there.