Home > Calculus > Formal Definition of a Limit at a Point

Is the statement "if f(c)=L, then the limit of f(x)=L as x approaches c" a true or false statement?

Answer 1

It is not true in general. That is: There are situations in which #f(c)=L#, but it is not true that #lim_(xrarrc)f(x) = L#.

Example 1

Define #f(x) = {(1/x,"if",x != 0),(1,"if", x=0):}#

#f(0) = 1#, but #lim_(xrarr0)f(x)# does not exist.

Example 2

Define #g(x) = {(x,"if",x != 0),(1,"if", x = 0):}#

#g(0)=1#, but #lim_(xrarr0)g(x) = 0#

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Answer 2

Emma Aldridge

True.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.