Limit of (ax - 1)/x

How to use the limit of Limit of (ax - 1)/x to solve the given limit with an exponential function: formula, 1 example, and its solution.

Formula

Formula

The limit of (ax - 1)/x as x → 0 is
ln a.

Limit of (ex - 1)/x

Example

Example

Solution

4x - 1
= 22x - 1
= (2x)2 - 12

Power of a Power

(2x)2 - 12 = (2x + 1)(2x - 1)

Factor the Difference of Two Squares: a2 - b2

Cancel the factors (2x + 1).

The exponent of 2x is x.
The denominator is also x.

So, as x → 0,
(2x - 1)/x → ln 2.

So ln 2 is the answer.