 It's helpful to keep in mind that the derivative is the slope of the tangent line. So, for example, we might have the graphs of y equals f of x and y equals g of x, and attempt to find the limit as x approaches 3 of the quotient. So, let's check our admission papers. The limit as x goes to 3 of f of x, we see that as x goes to 3, we see that the y values on y equals f of x go to 0, and so that means the limit will be 0, and as x goes to 3, the y values on y equals g of x also go to 0, and so the limit as x approaches 3 of g of x will be 0, and so L'Hopital's is relevant. And so the limit of the quotient of the functions is the limit of the quotient of the derivatives provided that everything exists. Now, remember the derivative is the slope of the line tangent to the graph, and so we take a look at our tangent lines. And so from the line tangent to the graph of y equals f of x at x equals 3, in other words, f' of 3, we see the slope is equal to 2, so f' of 3 is equal to 2. Similarly, we see that the graph of y equals g of x has a tangent line at x equals 3, and the slope of that tangent line is negative 1, and so g' of 3 is negative 1. And so our limit of the numerator is 2, our limit of the denominator is negative 1, and so the limit of the quotient is negative 2. And we can compose our functions. So let's say we have the graph of y equals f of x, and let's find the limit as x goes to 0 of this horrible mess. Now, since this is a graph of y equals f of x, we can read some information off the graph directly. So we're interested in what happens at x equals 0, and so we find f of 0 and f' of 0, negative 1 and 0. And this allows us to find the limit of the numerator and the limit of the denominator. And since both limits are 0, we can gain admission to the L'Hospital. So the limit of the quotient of the functions is equal to the limit of the quotient of the derivatives provided that everything exists, and so we'll find our derivatives of numerator and denominator, and we've already determined the value of f of 0 and f' of 0, so we'll substitute those in. Our numerator goes to 1, our denominator also goes to 1, and so the limit of the quotient is equal to 1 over 1 or 1. In some applications, we might only care about the sign of the limit, and so we see that as x approaches negative 2, f of x and g of x both go to 0, so we visit the L'Hospital, and the limit of the quotient of the functions is the limit of the quotient of the derivatives provided they all exist, and we see that the limit as x approaches negative 2 of f' of x and g, and that means we actually needed an extended L'Hospital stay. So let's differentiate again. Now we need to know something about the second derivative. So remember the second derivative algebraically corresponds to the geometric feature of the concavity of the graph. Now since y equals f of x is concave up, then our second derivative is going to be positive at x equals negative 2, and since the graph of y equals g of x is concave down, our second derivative will be negative at x equals negative 2, and so this quotient of the second derivatives will also be negative.