scientifico said:I've set up this unequation [itex]\frac{x-3}{x-1} < - M[/itex] to prove the limit using its definition but it doesn't lead to the result [itex]1<x<1+\frac{2}{M+1}[/itex]
A limit is a concept in calculus that describes the behavior of a function as the input values approach a specific point, called the limit point. It is denoted by the symbol "lim" and is defined as the value that the function approaches as the input values get closer and closer to the limit point.
The limit of a function can be calculated using the definition by evaluating the function at values closer and closer to the limit point and observing the trend of the output values. If the output values approach a specific value as the input values get closer to the limit point, then that value is considered as the limit of the function.
The definition of a limit provides a rigorous and precise way of determining the limit of a function. It allows us to formally prove the existence and value of a limit and provides a solid foundation for further mathematical analysis and applications.
The key components of the limit definition include the limit point, the function, and the limit value. The limit point is the point at which the input values are approaching, the function is the mathematical expression being evaluated, and the limit value is the value that the function approaches as the input values get closer to the limit point.
The limit definition is a more formal and rigorous approach to finding limits compared to other methods such as direct substitution or using algebraic manipulations. It provides a precise and unambiguous way of determining the limit of a function, whereas other methods may not always be applicable or may lead to incorrect results.