Limits part 2: Functions: Techniques
Substitution and one-sided limits
We just saw how we can use substitutions to calculate limits. In this paragraph we will consider one-sided limits.
Suppose that for some function we would like to calculate the value of
It makes sense to substitute and determine the limit for . A problem we encounter now is that approaches only from above so this notion belongs to . This means the limit unintentionally changes into a one-sided limit:
To ensure exists we should also determine . An option is to substitute in order to determine the left limit.
Look at
When we substitute this changes into
This is a standard limit with value .
In this example we have
in the substitution rule. The reason that is that approaches zero from above when approaches . When you substitute and you want to say something about the limit at zero, then you must explore the behaviour at both and .
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