# Functions: Limits

Using the definition of a limit (rather than the limit theorems) prove that

lim {x -> a+} f(x)

exists and find the limit in each of the following cases

a) f(x) = x/|x|, a = 0.

b) f(x) = x + |x|, a = -1.

c) f(x) = (x - 1)/(x^2 - 1), a = 1.

In which cases do

lim {x -> a-} f(x) and lim {x -> a} f(x)

also exist?

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#### Solution Preview

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First let me clearly state the definition of a limit in my words.

Definition: The statement lim {x -> a} f(x) has the following precise definition. Given any real number >0, there exists another real number >0 so that if 0<|x-a|<, then |f(x)-L|<.

Furthermore we can say the same for the two one sided limits.

Definition: The statement lim {x -> a+} f(x) has the following precise definition. Given any real number >0, there exists another real number >0 so that if 0<|x-a+|<, then |f(x)-L|<.

Definition: The statement lim {x -> a-} f(x) has the ...

#### Solution Summary

The existence of limits for functions is determined.