One-sided Limits

One-sided Limits

For some functions, it is appropriate to look at their behavior from one side only. If x approaches c from the right only, you write

  or if x approaches c from the left only, you write

It follows, then, that  if and only if 

Example 1: Evaluate 

Because x is approaching 0 from the right, it is always positive; is getting closer and closer to zero, so . Although substituting 0 for x would yield the same answer, the next example illustrates why this technique is not always appropriate.

Example 2: Evaluate .

Because x is approaching 0 from the left, it is always negative, and  does not exist. In this situation,  DNE. Also, note that  DNE because .

Example 3: Evaluate

  

a. As x approaches 2 from the left, x − 2 is negative, and | x − 2|=− ( x − 2); hence,

  

b. As x approaches 2 from the right, x − 2 is positive, and | x − 2|= x − 2; hence;

  

c. Because