Limits are written in the notation in the graphic. The sample in the graphic is read "the limit of f(x) as x approaches c." The most basic limit problems are for constants and polynomials. The limit of a constant, regardless of the value x approaches, is always the constant. No matter what x does, the constant will never change. For a polynomial, the limit is the function evaluated at the value x approaches.
As functions become more complex, they can be broken down using some properties of the limit operation. First, the limit of a sum is equal to the sum of the limits. Also, the limit of a product is equal to the product of the limits of the factors. These rules apply to subtraction and division as well and are illustrated in the graphic. The polynomial from the previous example proves the validity of these rules.
A limit implies that x never actually reaches the value it is defined to be approaching. So far, the solutions just happen to equal to the function evaluated at that value. If positive or negative infinity is the value x is approaching, the limit solution requires a little more intuition. If a denominator gets infinitely large, then the limit is zero. If the numerator gets infinitely large, then the limit is either positive or negative infinity.
Discontinuities in functions create a variety of special cases. The most common discontinuity occurs when the denominator of a function equals zero at the value x is approaching. Sometimes, you can avoid this situation by multiplying through the numerator and the denominator by a factor removes the discontinuity. Other times, an intuitive solution is necessary, as shown in the graphic. Another discontinuity occurs in certain types of periodic functions such as the tangent.
Limits provide a test for continuity of a function at a point. If a limit is taken from the left side of a point and then from the right side of the same point and they are the same, then the function is continuous at that point. If they are not equal, then the function has a discontinuity at that point. A limit from the left is noted by a minus (-) sign, and a limit from the right is noted with a plus (+) sign, as shown in the graphic.