Determine whether the function f(x) is continuous

[naspek]

Homework Statement

Given that

f(x) = { x + 1 ......; if x < 1
...{ 2 .....; if x = 1
...{ [4(x-1)] / (x^2 - 1) ; if x > 1

Determine whether the function f(x) is continuous at x = 1

i don't know how to start..
can someone give me an idea to start..

 

[VeeEight]

Try finding the limit of f as x approaches 1 from the left and the right (that is, with values of x < 1 and values of x > 1)

 

[naspek]

ok.. for

[tex]\lim_{x \to 1^{-}} f(x)[/tex]

i got 0

for

[tex]\lim_{x \to 1^{+}} f(x)[/tex]

i got 2

am i got it right?
so.. how can i conclude it?

 

[cap.r]

ok so now you have to go back to the calc one definition of continuity. the requirements were
If f is continuous at a then, these 3 facts have to hold
[tex] \lim_{x \to a^{-}} f(x) = f(a) [/tex]
[tex] \lim_{x \to a^{+}} f(x) = f(a) [/tex]
[tex] \lim_{x \to a} f(x) = f(a) [/tex]

 

[HallsofIvy]

ok.. for

[tex]\lim_{x \to 1^{-}} f(x)[/tex]

i got 0

For x< 0, f(x)= 1+ x. Are you saying that [itex]\lim_{x\to 0} 1+ x= 0[/itex]?

for

[tex]\lim_{x \to 1^{+}} f(x)[/tex]

i got 2

am i got it right?
so.. how can i conclude it?

This function is continuous at x=0 only if the limit there exists and is equal to f(0). The limit itself exist only if those two one sided limits are the same.