Limit of $\frac{3^{x} - 1}{x}$ as $x \to 0$

Q: What is the limit of (3^x - 1) / x as x approaches 0?

The limit of 3^x - 1x as x → 0 is ln 3.

The limit of $\frac{3^{x} - 1}{x}$ as $x \to 0$ is $\ln 3$ .

\lim_{x \to 0} \frac{3^{x} - 1}{x} = \ln 3

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Graph of (3^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 3 ≈ 1.0986123.

How to work it out

1
We want what ( $3^{x}$ - 1) / x approaches as x → 0.
$\lim_{x \to 0} \frac{3^{x} - 1}{x}$
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.
$\lim \frac{3^{x} - 1}{x} = \lim \frac{3^{x} \ln\left(3\right)}{1}$
4
Now substitute x = 0: the denominator is not 0 any more.
$\frac{1.0986123}{1} = 1.0986123$
5
Method: the derivative of $3^{x}$ at 0 is ln 3.
6
Result:
$\lim_{x \to 0} \frac{3^{x} - 1}{x} = \ln 3$

LaTeX source of the formulas on this page

\[ \lim_{x \to 0} \frac{3^{x} - 1}{x} = \ln 3 \]
\[ \lim_{x \to 0} \frac{3^{x} - 1}{x} \]
\[ \lim \frac{3^{x} - 1}{x} = \lim \frac{3^{x} \ln\left(3\right)}{1} \]
\[ \frac{1.0986123}{1} = 1.0986123 \]

Values of ( $3^{x}$ - 1) / x as x → 0 (computed)

x	f(x)	Distance from the limit
0.1	1.1612317	6.26e-2
-0.1	1.0404154	5.82e-2
0.01	1.1046692	6.06e-3
-0.01	1.0925996	6.01e-3
0.001	1.0992160	6.04e-4
-0.001	1.0980090	6.03e-4
0.0001	1.0986726	6.03e-5
-0.0001	1.0985519	6.03e-5
0.00001	1.0986183	6.03e-6
-0.00001	1.0986063	6.03e-6

Questions

What is the limit of (

3^{x}

- 1) / x as x approaches 0?

The limit of $\frac{3^{x} - 1}{x}$ as $x \to 0$ is $\ln 3$ .

Which form is it?

0/0.

Limit of 3x−1x\frac{3^{x} - 1}{x}x3x−1​ as x→0x \to 0x→0

How to work it out

Values of (3x3^{x}3x - 1) / x as x → 0 (computed)

Questions

More well-known limits

Limit of $\frac{3^{x} - 1}{x}$ as $x \to 0$

Values of ( $3^{x}$ - 1) / x as x → 0 (computed)