lucy4104 2 years ago is lim as x approaches c the same as f(c)?

1. ghazi

where is the function f(c)

2. lucy4104

c=3, f(x)= (x^2+5)/(x-6) continuous or no?... is the question.

3. ByteMe

at c=3, yes... the function is defined at x=3 so the limit of f is f(3)

4. lucy4104

what if x=c is not defined?

5. ByteMe

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.

6. lucy4104

the limit can still exist but not necessarily at f(c)?

7. ByteMe

yes... for example... |dw:1361168628388:dw| here, the limit of f(x) as x approaches c is a.... NOT f(c)

8. lucy4104

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?

9. ByteMe

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)

10. lucy4104

oh! so then if x=6 was not an asymptote, but a hole, the limit would be equivalent to f(c)??

11. ByteMe

***IF*** the function was CONTINUOUS at x=6, then the limit would be f(6)

12. ByteMe

seems like we're going in circles here but we're not... Continuity of a funtion is defined in terms of limits....

13. lucy4104

so...no?

14. ByteMe

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)...

15. lucy4104

yes!!!! ok, thank you!

16. ByteMe

yw...:)