Math Insight

Applet: Limit of b to the h minus one over h as h tends to zero converges to the natural logarithm

Demonstration that a function $$m(b)=\frac{b^h-1}{h}$$ approaches the natural logarithm in the limit that the parameter h goes to zero. For a given value of $h$, determined by the red slider, $(b^h-1)/h$ is plotted as a function of $b$ by the thin green curve. The graph of the function $\ln b$ is plotted by the thick blue curve. As you change $h$ to make it closer and closer to zero, the thin green curve converges to the thick blue curve, demonstrating that $(b^h-1)/h$ approaches $\ln b$ as $h$ approaches zero.

Applet file: limit_b_to_h_minus_1_over_h_ln.ggb

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