Math Insight

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

To estimate the limit $$\lim_{h \to 0} \frac{b^h-1}{h},$$ one can calculate $(b^h-1)/h$ for smaller and smaller values of $h$. The value of $b$ is determined by the number in the third column, which defaults to $b=2$. Enter a value for $h$ in the first column; the corresponding value of $(b^h-1)/h$ is displayed in the second column. To estimate the limit, enter smaller and smaller value of $h$. You should notice that, for sufficiently small values of $h$, the first few digits of $(b^h-1)/h$ remain unchanged as you make $h$ even smaller. In that case, those unchanged digits represent a good estimate of the limit, accurate to the number of digits than remain unchanged. One should obtain the same limiting value if one uses negative values for $h$ and enters successive values of $h$ that are smaller and smaller in absolute value (i.e., closer to zero).

Applet file: limit_b_to_h_minus_1_over_h.ggb

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