Determine whether k(x) is continuous at x=1.

Question

k(x) =1-x², x<1

k(x)= x+1, 1<= x<2

k(x) = √x+2, x>=2

Answer 1

For k(x) to be continuous at x = 1, the limit of k(x) must exist at x = 1 and be equal to k(1).

The limit of k(x) as x approaches 1 form the left is 1 - (1)² = 0, but the limit of k(x) as x approaches 1 from the right is 1+1 = 2. So, since the one-sided limits are not the same, the limit of k(x) as x approaches 1 does not exist.

Therefore, k(x) is NOT continuous at x = 1.

If you graph the function, you will see that there is a gap in the graph when x = 1. Continuous function graphs do not have gaps or holes. .