The computations of the expected value and variance of an exponential random variable relied on the fact that, for any positive number k, be^-kb and b2 e^-kb approach 0 as b gets large. That is,

The validity of these limits for the case k = 1 is shown in Figs. 8 and 9. Convince yourself that these limits hold for all positive values of k by generating graphs for k = .1, k = .5, and k = 2.

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X lim x→∞ekx = 0 and x² lim x-xekx = 0.