Limit of $\frac{10^{x} - 1}{x}$ as $x \to 0$

How to work it out

1. 1

   We want what ($10^{x}$ - 1) / x approaches as x → 0.

   $$\lim_{x \to 0} \frac{10^{x} - 1}{x}$$
2. 2

   Direct substitution gives an indeterminate form: 0/0. It does not tell us the limit.

3. 3

   L'Hôpital's rule: when the form is 0/0, the limit equals the limit of the ratio of the derivatives.

   $$\lim \frac{10^{x} - 1}{x} = \lim \frac{10^{x} \ln\left(10\right)}{1}$$
4. 4

   Now substitute x = 0: the denominator is not 0 any more.

   $$\frac{2.3025851}{1} = 2.3025851$$
5. 5

   Method: the derivative of $10^{x}$ at 0 is ln 10.

6. 6

   Result:

   $$\lim_{x \to 0} \frac{10^{x} - 1}{x} = \ln 10$$

Values of ($10^{x}$ - 1) / x as x → 0 (computed)

Questions

More well-known limits

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