What is the limit and how do you get it?

Question

Estimate the limit as x approaches 0 of ((5^x)-1) / x

answer to the nearest 0.001

Answer 1

Answer: 1.609, i.e., ln 5 (the natural log of 5)

How to get it:

Use L'Hospital's Rule (LHR), which allows you to form the ratio of the derivative of the numerator (N) to the derivative of the denominator (D), both with respect to x, to evaluate the limit in this case.

Here, N = (5^x) - 1 and D = x.

Their derivatives are dN/dx = (5^x)*ln(5) and dD/dx = 1

The ratio is (5^x)*ln(5)/1.

In the limit, as x nears 0, 5^x approaches 1 (i.e., 5^0 = 1).

Therefore, by means of LHR, the limit = ln(5) = about 1.609.