3 reasons Limits do NOT exist
1) Discontinuity: One reason a limit may not exist is if there is a discontinuity in the function at the point in consideration
1) Discontinuity: One reason a limit may not exist is if there is a discontinuity in the function at the point in consideration. A function can have a jump, a hole, or an asymptote, and in such cases, the limit will not exist. For example, consider the function f(x) = 1/x. As x approaches 0, the function approaches positive or negative infinity depending on the direction of approach, so the limit does not exist at x = 0.
2) Oscillation: Another reason a limit may not exist is if the function oscillates infinitely or behaves erratically as x approaches a certain value. In such cases, the limit does not approach a single value or it may not approach any value at all. For instance, consider the function g(x) = sin(1/x). As x approaches 0, the value of sin(1/x) oscillates indefinitely between -1 and 1, preventing the existence of a limit.
3) Undefined or Indeterminate Forms: Sometimes, limits do not exist because they fall into undefined or indeterminate forms. These are situations where the expression does not have a well-defined value. For example, if we have the expression 0/0, this is an undefined form, and no limit can be determined directly from it. Similarly, expressions like ∞/∞ or 0 * ∞ are indeterminate forms, and determining their limits may require more advanced techniques like L’Hôpital’s rule.
It is important to note that the non-existence of a limit does not imply that the function is not defined or that the value cannot be approximated. It simply means that the limit does not exist at that particular point.
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