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Is the following function continuous at #x=3# ?

#f(x) = { (2, " if " x=3), (x-1, " if " x > 3), ((x+3)/3, " if " x < 3) :}#

Answer 1

Cameron Alexander

Yes

Given:

#f(x) = { (2, " if " x=3), (x-1, " if " x > 3), ((x+3)/3, " if " x < 3) :}#

We find:

#lim_(x->3-) f(x) = lim_(x->3-) (x+3)/3 = (color(blue)(3)+3)/3 = 2#

#lim_(x->3+) f(x) = lim_(x->3+) (x-1) = color(blue)(3) - 1 = 2#

#f(3) = 2#

So the left and right limits agree and are equal to #f(3)#.

So this #f(x)# is continuous at #x=3#

graph{((x-3)/abs(x-3)+1)/2(x-1)+(1-(x-3)/abs(x-3))/2((x+3)/3) [-2.955, 7.045, -0.5, 4.5]}

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Answer 2

Isabella Ackerman

To determine if the function is continuous at ( x = 3 ), we need to check three conditions:

The function must be defined at ( x = 3 ).
The limit of the function as ( x ) approaches 3 must exist.
The limit of the function as ( x ) approaches 3 must equal the value of the function at ( x = 3 ).

Without knowing the specific function, it's not possible to provide a definitive answer. You would need to provide the function itself or additional context for a more accurate assessment of its continuity at ( x = 3 ).

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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.