The point of discontinuity can be easily detected when you see the face of the graph. If it contains any breaks, there are points of discontinuity. You can determine this analytically if when you substitute x to the equation, the answer would be undefined. Undefined numbers are ∞-∞, 0/0, 0×∞, 0^0, 1^∞ or any number divided by zero.
For the expression f(x)=x^2+5x+6/x+3, you see that there is a term 6/x. Therefore, the expression can be continuous when x=0 because 6/0 or any number divided by zero is undefined. To verify this, let's see the graph by assigning values of x and plotting them against their values of y. As you can see, the curves only approach zero but never touches zero. These are called asymptotes.
