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

1. As x approaches 2 from the left, x − 2 is negative, and | x − 2|=− ( x − 2); hence,
   
   
2. As x approaches 2 from the right, x − 2 is positive, and | x − 2|= x − 2; hence;
   
   
3. Because

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By Muhammad Atif Saeed on 01:45. Filed under Calculus , Mathandstat . Follow any responses to the RSS 2.0. Leave a response