Indicate if the function fxx21 if x 3 2x4 if x3 is continuou

Indicate if the function f(x)={x^2+1 if x 3 2x+4 if x>3 is continuous at x=3

Solution

lim_x->3^- f(x)

=lim_x->3^- (x^2+1)

=3^2+1

=9+1

=10

f(3)=3^2+1

=9+1

=10

lim_x->3⁺ f(x)

=lim_x->3⁺ (2x+4)

=(2*3 +4)

=6+4

=10

lim_x->3^- f(x)=f(3)=lim_x->3⁺ f(x)

so the function f(x)={x^2+1 if x 3 2x+4 if x>3 is continuous at x=3