Function question - is a certain function continuous at a certain point (x=1)?

Question

The question states the "The function is f(x) = (x²+x-2)/(x-1) if x > 1 and = x²+2 if x <= 1. Is the function continuous at x = 1?"

The question is then followed by "Justify your argument by using limit calculations".

I am at a complete dead end for how to solve this, is there a reasonably simple explanation for how to solve this?

Answer 1

,as x approaches 1 from the greater than one side, the limit is 3,

just divide f(x) numerator by (x-1) and f(x)=(x+2) for x>1

as x approaches 1 from the less than side the limit is 3 ,,,,f(x)=x²+2

for continuity, need f(x=1) equal to both approaching limits (which need

to match also) since f(x=1)=1²+2=3 the function is continuous.

This is a case where f(x) is continuous at x=1, but the derivative will be

discontinuous at x=1.

Answer 2

If you factor the numerator, you will see that you have a removable discontinuity for x>1.

lim_x→1+ (x²+x-2)/(x-1) = lim_x→1+(x+2) = ?

lim_x→1-(x²+1) = ?

If both the left and right limits are equal, then the function is continuous at x=1. If the limit aren't equal then the function is not continuous at x=1.