anonymous
  • anonymous
is lim as x approaches c the same as f(c)?
Mathematics
schrodinger
  • schrodinger
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ghazi
  • ghazi
where is the function f(c)
anonymous
  • anonymous
c=3, f(x)= (x^2+5)/(x-6) continuous or no?... is the question.
anonymous
  • anonymous
at c=3, yes... the function is defined at x=3 so the limit of f is f(3)

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anonymous
  • anonymous
what if x=c is not defined?
anonymous
  • anonymous
if f(x) is not defined at x=c, then the limit can still exist but not necessarily at f(c). also, if there is a vertical asymptote at x=c, then the limit does not exist.
anonymous
  • anonymous
the limit can still exist but not necessarily at f(c)?
anonymous
  • anonymous
yes... for example... |dw:1361168628388:dw| here, the limit of f(x) as x approaches c is a.... NOT f(c)
anonymous
  • anonymous
OHHH so then in that case it would be dicontinuous? When can you tell, or what can you do, to know that something is continuous, but algebraically? step by step and explain?
anonymous
  • anonymous
yes... but in your original function, it is continuous at all x values except at x=6. so in your problem, the limit of f as x approaches c for any value OTHER THAN 6, will be f(c)
anonymous
  • anonymous
oh! so then if x=6 was not an asymptote, but a hole, the limit would be equivalent to f(c)??
anonymous
  • anonymous
***IF*** the function was CONTINUOUS at x=6, then the limit would be f(6)
anonymous
  • anonymous
seems like we're going in circles here but we're not... Continuity of a funtion is defined in terms of limits....
anonymous
  • anonymous
so...no?
anonymous
  • anonymous
if the "hole" you're referring to is a REMOVABLE discontinuity, then yes, the limit would be f(6) or as you said, f(c)...
anonymous
  • anonymous
yes!!!! ok, thank you!
anonymous
  • anonymous
yw...:)

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