 So let's put some of our ideas together and see what we can determine about a graph from the properties of a function. So suppose f of x is a function with all of the following properties. f of x equals 0 has solutions x equals 8 and x equals negative 4 and no others. The limit as x goes to infinity of f of x is 3 and the limit as x goes to minus infinity of f of x is also equal to 3. The limit as x approaches 3 from below of f of x is equal to 10. The limit as x approaches 5 from above of f of x is minus infinity and the limit as x approaches 5 from below of f of x is positive infinity and f of x is continuous as long as x is not equal to 5. Let's start out by finding the limit as x approaches 3 from above of f of x and importantly we'll defend our answer by explicitly referring to the given properties of f of x. So of this information the most general is this fact that f of x is continuous for x not equal to 5 and so we might expect this to be the most useful piece of information and since definitions are the whole of mathematics we know the definition of continuity. But in case we need a reminder or something to refer to let's pull in that definition and the key concept of continuity is that the limit is equal to the function value. So we know that f of x is continuous for x not equal to 5 and that means we know that the limit as x approaches 3 of f of x must exist. Now since we're trying to find a one-sided limit we should remember that the unqualified limit exists if and only if the one-sided limits exist and they agree. And so that means the limit as x approaches 3 from above must be the same as the limit as x approaches 3 from below. Fortunately we know the limit as x approaches 3 from below it's equal to 10 and so that tells us the limit as x approaches 3 from above must also be equal to 10. One important note on this problem it's not enough for this question to simply give the answer the limit is equal to 10. You want to defend the answer by explicitly referring to the given properties of f of x. In this case the important properties are the fact that f of x is continuous which means that the limit exists which means that the one-sided limits have to agree and all of these are important components of the complete answer which means that all of these need to be included in an answer to this question. How about the limit as x approaches 3 of f of x? Again since we're all good calculus students we know what the definition of continuity is which again requires that the limit exists and the limit only exists if the one-sided limits exist and they agree. We use this to determine the value of the limit as x approaches 3 from above but let's take it to the next level. From the definition of one-sided limits we know that if the one-sided limits agree then the limit itself is the common value. So we know the limit as x approaches 3 of f of x is equal to the limit as x approaches 3 from above of f of x which is equal to the limit as x approaches 3 from below of f of x. And so the limit as x approaches 3 is equal to the limit as x approaches 3 from below and we're given this limit. How about f of 3? Well again we know our definition of continuity and the key point about continuity is that the function value is equal to the limit. Since f of x is continuous at x equals 3 we know that f of 3 is the limit as x approaches 3 of f of x and we've already determined what that is. How about the sine positive or negative of f of 0? We know f of x equals 0 at x equals 8 and x equals negative 4 so the graph passes through 8, 0 and negative 4, 0. We also know that f of 3 is equal to 10 so the graph passes through the point 3, 10. Now the obvious way to go from negative 4, 0 to 3, 10 will cut through the positive portion of the y-axis and that'll make f of 0 positive and it's important in life to remember that you always take the first answer that you're given or maybe not. I want to believe but let's have a reason to believe. Could the graph pass through the negative part of the y-axis? Since the function is continuous everywhere except at x equal to 5 then we know that however we join these two points it will be through some continuous curve. If we do go through the negative portion of the y-axis on our way to the point 3, 10 we see that we're going to have to cross the x-axis at some additional point. But one of the things we know is that f of x equals 0 only at 8 and at negative 4 and so this additional point can't exist which means we can't take this downward path. We have to take the path that passes through the positive portion of the y-axis and so we might say the following if f of 0 is negative the graph y equals f of x must pass through a third point on the x-axis to reach 3, 10 but this doesn't happen so f of 0 must be positive and this allows us to sketch a graph of y equals f of x that shows all of the preceding properties. So first, we know that the graph passes through the points 8, 0 negative 4, 0 and 3, 10. We also know that f of 0 is positive so we cross the y-axis at some point above the x-axis. We also know the graph is discontinuous at x equals 5 so we'll draw a do not cross line across x equals 5 as x goes to infinity f of x goes to 3 so that means that y is going to go to 3 so let's put in a destination line at y equals 3 which means as we move to the right the height above the x-axis approaches 3 since we know the point 8, 0 is on the graph let's move to the right from this point allowing our height above the x-axis to get close to 3 as x goes to minus infinity f of x goes to 3 as we move left our y-values approach 3 and since this point negative 4, 0 is on the graph we'll start there and move left allowing our y-values to get close to 3 as x approaches 5 from above f of x goes to minus infinity so our y-values go crashing downward and so we might start at this point 8, 0 we'll move towards x equal to 5 allowing our y-values to crash as x goes to 5 from below f of x goes to positive infinity and so do our y-values so we might start at 3, 10 and move towards x equals 5 allowing our y-values to get higher and higher finally since the graph is continuous we'll need to join these two segments together through the point on the y-axis and so our final graph might look something like this