### 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

#### Applet links

This applet is found in the pages

#### General information about Geogebra Web applets

This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.

You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.