Limit of $\frac{5^{x} - 1}{x}$ as $x \to 0$

Q: What is the limit of (5^x - 1) / x as x approaches 0?

The limit of 5^x - 1x as x → 0 is ln 5.

The limit of $\frac{5^{x} - 1}{x}$ as $x \to 0$ is $\ln 5$ .

\lim_{x \to 0} \frac{5^{x} - 1}{x} = \ln 5

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Graph of (5^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 5 ≈ 1.6094379.

How to work it out

1
We want what ( $5^{x}$ - 1) / x approaches as x → 0.
$\lim_{x \to 0} \frac{5^{x} - 1}{x}$
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.
$\lim \frac{5^{x} - 1}{x} = \lim \frac{5^{x} \ln\left(5\right)}{1}$
4
Now substitute x = 0: the denominator is not 0 any more.
$\frac{1.6094379}{1} = 1.6094379$
5
Method: the derivative of $5^{x}$ at 0 is ln 5.
6
Result:
$\lim_{x \to 0} \frac{5^{x} - 1}{x} = \ln 5$

LaTeX source of the formulas on this page

\[ \lim_{x \to 0} \frac{5^{x} - 1}{x} = \ln 5 \]
\[ \lim_{x \to 0} \frac{5^{x} - 1}{x} \]
\[ \lim \frac{5^{x} - 1}{x} = \lim \frac{5^{x} \ln\left(5\right)}{1} \]
\[ \frac{1.6094379}{1} = 1.6094379 \]

Values of ( $5^{x}$ - 1) / x as x → 0 (computed)

x	f(x)	Distance from the limit
0.1	1.7461894	1.37e-1
-0.1	1.4866008	1.23e-1
0.01	1.6224591	1.30e-2
-0.01	1.5965557	1.29e-2
0.001	1.6107338	1.30e-3
-0.001	1.6081435	1.29e-3
0.0001	1.6095674	1.30e-4
-0.0001	1.6093084	1.30e-4
0.00001	1.6094509	1.30e-5
-0.00001	1.6094250	1.30e-5

Questions

What is the limit of (

5^{x}

- 1) / x as x approaches 0?

The limit of $\frac{5^{x} - 1}{x}$ as $x \to 0$ is $\ln 5$ .

Which form is it?

0/0.

Limit of 5x−1x\frac{5^{x} - 1}{x}x5x−1​ as x→0x \to 0x→0

How to work it out

Values of (5x5^{x}5x - 1) / x as x → 0 (computed)

Questions

More well-known limits

Limit of $\frac{5^{x} - 1}{x}$ as $x \to 0$

Values of ( $5^{x}$ - 1) / x as x → 0 (computed)