Inﬁnite Discontinuities In an inﬁnite discontinuity, the left- and right-hand limits are inﬁnite; they may be both positive, both negative, or one positive and one negative. y x Figure 1: An example of an inﬁnite discontinuity: 1 x 1 1 = ∞ and lim− = −∞. Saying that a x x→0 x limit is ∞ is diﬀerent from saying that the limit doesn’t exist – the values of x1 are changing in a very deﬁnite way as x → 0 from either side. (Note that it’s 1 not true that lim = ∞ because ∞ and −∞ are diﬀerent.) x→0 x There are two more things we can learn from this example. First, sketch the d 1 1 graph of dx x = − x2 ; it also has an inﬁnite discontinuity at x = 0. Notice that the derivative of the function x1 is always negative. It may seem strange to you that the derivative is decreasing as x approaches 0 from the positive side while 1 x is increasing, but very often the graph of the derivative will look nothing like the graph of the original function. What the graph of the derivative − x12 is showing you is the slope of the graph of x1 . Where the graph of x1 is not very steep, the graph of − x12 lies close to the x-axis. Where the graph of x1 is steep, the graph of − x12 is far away from the x-axis. The value of − x12 is always negative, and the graph of x1 always slopes downward. Finally, x1 is an odd function and − x12 is an even function. When you take the derivative of an odd function you always get an even function and vice-versa. If you can easily identify odd and even functions, this is a good way to check From Figure 1, we see that lim+ x→0 1 y x y’ Figure 2: Top: graph of f (x) = x 1 1 and Bottom: graph of f � (x) = − 2 x x your work. 2