What Is a Limit in Precalculus? 

A limit is a mathematical concept that describes the behavior of a function as it approaches a particular value. The most obvious application of a limit is to determine the height of a function at a given point. However, limits can also be found for functions with varying degrees of continuity. In fact, they are crucial to many different mathematical analyses. They can be described using algebra, and even demonstrated graphically. 

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The most important thing to know about a limit is that it only exists when both the left and right sides of the function meet at a given height. This doesn’t mean that both sides of the function are equal, however. It does mean that they are at least in the same direction, which is a step in the right direction. Limits can be defined for either a real number or a complex fraction. 

The best way to determine the magnitude of a limit is to evaluate the function at each input value and then evaluate it at its output values. Then, you can determine the height of the function in the context of its surrounding neighborhood. You can use the graphing utility of a calculator to estimate the size of the limit. 

A well-behaved function has a limit as x approaches zero. This is especially true of rational functions with a non-zero denominator. The same goes for functions with a rational root. Similarly, a rational function with an equal power of x has a limit as x approaches infinity. 

While a function can be evaluated in an infinite number of ways, there are a few standardized methods that will give you a solid footing. Those methods include the axiom, the LCD, and the piecewise evaluation. 

A good example of a functional limit is the difference between the sum of two quotients. Using the LCD method, the difference can be written as a single quotient. Likewise, a rational function that is rational at all points in the interval can be simplified by multiplying the numerator by the least common denominator, which will give you a function that can be evaluated. Similarly, polynomials with a degree of three aren’t as amenable to this type of solution. 

Finding the one-sided limit is a bit more difficult. You can do this by examining the graph of a function, though the limit may not be apparent if the graph is not continuous. On the other hand, the sum of individual terms in a polynomial function can be derived by factoring. 

Limits are also used to describe the behavior of a function as it approaches the x-axis. This is called a limit around a point, and will probably be easier to figure out if you are familiar with a good graphing utility. 

Another trick of the trade is to find the smallest number of steps that will get you halfway to a certain destination. Taking small steps and seeing how far you can go is a great head start for students new to calculus.