Limit of $\frac{2^{x} - 1}{x}$ as $x \to 0$

Q: What is the limit of (2^x - 1) / x as x approaches 0?

The limit of 2^x - 1x as x → 0 is ln 2.

The limit of $\frac{2^{x} - 1}{x}$ as $x \to 0$ is $\ln 2$ .

\lim_{x \to 0} \frac{2^{x} - 1}{x} = \ln 2

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Graph of (2^x - 1) / x near x = 0. The open circle marks the point x = 0, where the function may be undefined; the curve still heads to ln 2 ≈ 0.69314718.

How to work it out

1
We want what ( $2^{x}$ - 1) / x approaches as x → 0.
$\lim_{x \to 0} \frac{2^{x} - 1}{x}$
2
Direct substitution gives an indeterminate form: 0/0. It does not tell us the limit.
3
L'Hôpital's rule: when the form is 0/0, the limit equals the limit of the ratio of the derivatives.
$\lim \frac{2^{x} - 1}{x} = \lim \frac{2^{x} \ln\left(2\right)}{1}$
4
Now substitute x = 0: the denominator is not 0 any more.
$\frac{0.69314718}{1} = 0.69314718$
5
Method: the derivative of $2^{x}$ at 0 is ln 2.
6
Result:
$\lim_{x \to 0} \frac{2^{x} - 1}{x} = \ln 2$

LaTeX source of the formulas on this page

\[ \lim_{x \to 0} \frac{2^{x} - 1}{x} = \ln 2 \]
\[ \lim_{x \to 0} \frac{2^{x} - 1}{x} \]
\[ \lim \frac{2^{x} - 1}{x} = \lim \frac{2^{x} \ln\left(2\right)}{1} \]
\[ \frac{0.69314718}{1} = 0.69314718 \]

Values of ( $2^{x}$ - 1) / x as x → 0 (computed)

x	f(x)	Distance from the limit
0.1	0.71773463	2.46e-2
-0.1	0.66967008	2.35e-2
0.01	0.69555501	2.41e-3
-0.01	0.69075046	2.40e-3
0.001	0.69338746	2.40e-4
-0.001	0.69290701	2.40e-4
0.0001	0.69317120	2.40e-5
-0.0001	0.69312316	2.40e-5
0.00001	0.69314958	2.40e-6
-0.00001	0.69314478	2.40e-6

Questions

What is the limit of (

2^{x}

- 1) / x as x approaches 0?

The limit of $\frac{2^{x} - 1}{x}$ as $x \to 0$ is $\ln 2$ .

Which form is it?

0/0.

Limit of 2x−1x\frac{2^{x} - 1}{x}x2x−1​ as x→0x \to 0x→0

How to work it out

Values of (2x2^{x}2x - 1) / x as x → 0 (computed)

Questions

More well-known limits

Limit of $\frac{2^{x} - 1}{x}$ as $x \to 0$

Values of ( $2^{x}$ - 1) / x as x → 0 (computed)