Show that the limit as h approaches 0 of (e^h-1 ) all over h = ln e = 1 using at least two numerical examples.
To show that the limit as h approaches 0 of (e^h-1)/h is equal to ln e = 1, we can use numerical examples to demonstrate the behavior of the expression as h gets closer to 0.
Let's calculate the limit for two different values of h: h = 0.1 and h = 0.01.
Example 1: h = 0.1
We need to find the value of (e^h - 1)/h as h approaches 0.1.
(e^0.1 - 1) / 0.1 ≈ (1.10517092 - 1) / 0.1 ≈ 0.10517092 / 0.1 = 1.0517092
Example 2: h = 0.01
We need to find the value of (e^h - 1)/h as h approaches 0.01.
(e^0.01 - 1) / 0.01 ≈ (1.01005017 - 1) / 0.01 ≈ 0.01005017 / 0.01 = 1.005017
As we can observe, as h gets closer to 0, the value of (e^h - 1)/h approaches 1. These numerical examples provide evidence to support the assertion that the limit as h approaches 0 is 1.
In order to get more accurate results, you can use a calculator or a programming language to perform the calculations. Simply substitute the value of h into the expression (e^h - 1)/h and evaluate the result. Repeat this process for smaller and smaller values of h to observe the trend towards 1.