f$($x$)=$

$$cx$2+9$x$$if$$ x $<4$x$3$ cxif$$ x $4$

We note that for all values of the constant c the function f is continuous on both

$($$,4)$

and

$(4,$ $).$

The only case that we need to consider is when

x $=4.$

For the function f to be continuous on

$($$,$ $),$

we need to ensure that there is not a discontinuity at

x $=4.$

To do so$,$ we must determine the value of c such that the following holds.

$$lim$$x$4$f$($x$)=$lim$$x$4+$f$($x$)$

Find the following limits$.$

lim$$x$4$f$($x$)=$lim$$x$4$