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 of (e^h - 1)/h as h approaches 0 is equal to ln(e) = 1, we can evaluate the expression for (e^h - 1)/h at two different values of h that are very close to 0. Let's choose h = 0.1 and h = 0.01.
Step 1: Calculate the value of the expression at h = 0.1:
(e^h - 1)/h = (e^0.1 - 1)/0.1
Using a calculator, we find that e^0.1 ≈ 1.10517
Substituting the values:
(1.10517 - 1)/0.1 ≈ 0.0517
Step 2: Calculate the value of the expression at h = 0.01:
(e^h - 1)/h = (e^0.01 - 1)/0.01
Using a calculator, we find that e^0.01 ≈ 1.01005
Substituting the values:
(1.01005 - 1)/0.01 ≈ 0.01005
Step 3: Compare the results:
As h approaches 0, the value of (e^h - 1)/h approaches ln(e) = 1.
From our calculations, we can observe that as h gets smaller (0.1 to 0.01), the value of (e^h - 1)/h gets closer to 1. This provides evidence that the limit of the expression as h approaches 0 is indeed 1.
Therefore, we have shown the limit as h approaches 0 of (e^h - 1)/h = ln(e) = 1 using two numerical examples.