 So let's consider a few more cases of the derivative. So it's helpful to remember that on the graph of y equals f of x, the difference quotient is going to be the slope of the line between x equals a, and x equals a plus h on the graph. So let's see if we can find the limit as h goes to 0 from above of the difference quotient, as h goes to 0 from below of the difference quotient, and then the limit as h goes to 0. So in this case, the difference quotient itself is going to be the slope of the secant line between x equals 3 and x equals 3 plus h. Now since we want the limit as h goes to 0 from above, then x equals 3 plus h will be to the right of x equals 3. And so here we have our point at x equals 3, and someplace off to the right we have our point at x equals 3 plus h. And we'll draw the secant line. And now we're going to let h go to 0, and that means this point on the right is going to come closer to the point at x equals 3. And if we do that, it appears that as h goes to 0 from above, the slope of the secant line will be positive. Again, the difference quotient itself will be the slope of the secant line between two points, but since we want the limit as h goes to 0 from below, then x equals 3 plus h will be to the left of x equals 3. So we have our point at x equals 3, a point somewhere to the left, the secant line, and as the point gets close, we see that it appears that as h goes to 0 from below, the slope of the secant line will be negative. Now because the two limits disagree, a positive number can't be equal to a negative number. This means that the limit itself doesn't exist. Now remember the derivative is an algebraic statement, but it also has to do with the geometry of the graph. And note that the graph has a corner at x equal to 3. This is called a cusp. What about a graph like this? So notice that the point is actually at x equal to 3. So we can try to find the left and right limits. So again the difference quotient will be the slope of the secant line between x equals 3 and x equals 3 plus h, and since we want the limit as h goes to 0 from above, then x equals 3 plus h will be to the right of x equals 3. And as we let h go to 0, it appears that as h goes to 0 from above, the secant line becomes more and more horizontal, and so the slope of the secant line tends to 0. And so we might say that our limit as h goes to 0 from above is going to be 0. Again the difference quotient is the slope of the secant line between x equals 3 and x equals 3 plus h. Since we want the limit as h goes to 0 from below, then x equal to 3 plus h will actually be to the left of x equals 3. And we see that as h goes to 0 from below, the secant line will be more and more vertical, and so the slope of the secant line tends to infinity. And again this disagreement between the limit from the left and from the right means that the limit itself doesn't exist. And this time notice the graph is discontinuous at x equal to 3. And this leads to the following result, which is an important consequence of the derivative. The derivative is undefined wherever the graph has a cusp or a discontinuity.