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How to Read Limits in Calculus Equations

In calculus, taking the limit of a function means finding the value of the function as its variable "x" approaches a specified number "a." Generally, the limit of a function is equal the function at "a," found by direct substitution. However, in the case of rational functions, logarithms and other functions with undefined values, the limit cannot be determined by direct substitution. Usually, a function has a limit at all values of "a." But sometimes there is no limit at "a," such as when a graph goes to infinity. Other times, the limit can vary depending on the direction "x" is approaching "a."

Instructions

- 1
  Identify the components of the limit symbology and understand their function. Look at the general limit notation: lim (x -> a) f(x). Pronounce the symbols as, "the limit of f of x as x approaches a."
- 2
  Substitute "a" into f(x) to see if the function is solvable at "a." If it is solvable, then the limit of the function equals the value of "a." For example, substituting "a" into the function for the limit, lim (x -> 2) x^2 becomes: (2)^2 = 4. So, the limit as "x" approaches "a" for this function equals 4.
- 3
  Substitute values of "x" from the "left" of "a" into the function. Values of "x" can be arbitrarily close to the value of "a" but never equal to "a." For example, substituting values from the left of a = 2 for the limit, lim (x -> 2) x^2 finds: (0)^2 = 2; (1)^2 = 1, (1.5)^2 = 2.25, (1.9)^2 = 3.61, (1.999)^2 = 3.996. As the value of x becomes closer to a = 2, the value of f(x) appears to become closer and closer to 4.
- 4
  Substitute values of "x" from the "right" of "a" into the function. Values of "x" can be arbitrarily close to the value of a but never equal to "a." For example, substituting values from the right of a = 2 for the limit, lim (x -> 2) x^2 finds: (4)^2 = 16; (3)^2 = 9, (2.5)^2 = 6.25, (2.1)^2 = 4.41, (2.001)^2 = 4.004. As the value of x becomes closer to a = 2, the value of f(x) appears to become closer and closer to 4.
- 5
  Look at the limits from each side of "a" and determine whether or not they are equal. If so, then the limit for the functions exists and is equivalent to the value of "a." If the two limits are not equal then the limit for x = a does not exist. Instead, there are two limits, called one-sided limits, for the function: the limit "from the right" and the limit "from the left" of "a."

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