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

How to work it out

1. 1

   We want what ($x^{5}$ - 1) / (x - 1) approaches as x → 1.

   $$\lim_{x \to 1} \frac{x^{5} - 1}{x - 1}$$
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.

   $$\lim \frac{x^{5} - 1}{x - 1} = \lim \frac{5 x^{4}}{1}$$
4. 4

   Now substitute x = 1: the denominator is not 0 any more.

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

   Method: this is the derivative of $x^{5}$ at 1, namely 5.

6. 6

   Result:

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

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

Questions

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