The limit of a function or a sequence in mathematics is the number the function or sequence approaches as the independent variable (x) approaches a particular number. The definition is as follows: If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L. This is often expressed lim x c f(x) = L.
There are three major areas where the limit of a function or sequence does not exist. In order for a function or sequence to have a limit, the function or sequence must be approaching the said value of the function or sequence from the left and right sides of x. Left and right hand behavior must agree. If it does not, the function does not have a limit. The behavior of the function or sequence cannot approach positive or negative infinity from the left or right of the x value in question. If this is the case, the limit does not exist as x approaches c. This is considered unbounded behavior. The third way a limit will not exist for a function or sequence is when the value of the function or sequence oscillates between two fixed values as x approaches c.
The limit value may not be a value of the function, it is simply the value the function approaches. It is often necessary to simplify a function before ruling out the lack of a limit at a particular point. It is possible to evaluate a limit numerically (e.g., looking at a table), graphically (analyzing a graph at a particular value of x), or analytically (solving the limit algebraically). This might mean simply plugging the x value into the function and simplifying, or by simplifying the expression and then evaluating at the given value of x.
If the limit of a function does not exist because the left and right hand behavior do not agree, it is possible to talk about one-sided limits. This is when the behavior of a function is examined as x approaches c from only one side. If the function is a step function, it is possible to obtain one limit value as x is approached from the left and a completely different limit value when it is approached from the right. The limit at the x as it approaches c does not exist, but the one-sided limit does exist.
Limits play a key role in the development of the derivative in calculus. A derivative represents the slope of the line tangent to a curve at any given point. Using the limit process, it is possible to find the exact slope of the tangent line at a given point. The concept of the limit minimizes the distance between the point on the graph in question and a second point used to calculate the slope. As the distance between the two points collapses, the slope of the function at a given point is identified.
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