continuity calculas 1

Question

The function  is given by the formula    f(x)=(5x³+16²+10x+21)/x+3     , when x<-3 

 and by the formula f(x)=-3x²+2x+a     , when x>= -3.

What value must be chosen for  in order to make this function continuous at -3?

a = 

Answer 1

A function f is continuous at the point x = a if the limits of the function as it approaches the point from the right and the left are equal. In equation form this looks like:

lim_x->a- f(x) = lim_x->a+ f(x)

In this case the point we are looking at is x = -3. The function we are given is a piecewise function with a different formula on either side of x = -3. Thus, we must be careful to use the correct formula when taking the limits from the left and right. Plugging in the correct formulas gives:

lim_x->-3^- (5x³+16²+10x+21)/x+3 = lim_x->-3⁺ f(x)=-3x²+2x+a

If you need help with these limits, I'll be checking in.