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

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

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

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

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

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

How to work it out

1
We want what ( $x^{2}$ - 1) / (x - 1) approaches as x → 1.
$\lim_{x \to 1} \frac{x^{2} - 1}{x - 1}$
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.
$\lim \frac{x^{2} - 1}{x - 1} = \lim \frac{2 x}{1}$
4
Now substitute x = 1: the denominator is not 0 any more.
$\frac{2}{1} = 2$
5
Method: factor x² − 1 = (x − 1)(x + 1) and cancel.
6
Result:
$\lim_{x \to 1} \frac{x^{2} - 1}{x - 1} = 2$

LaTeX source of the formulas on this page

\[ \lim_{x \to 1} \frac{x^{2} - 1}{x - 1} = 2 \]
\[ \lim_{x \to 1} \frac{x^{2} - 1}{x - 1} \]
\[ \lim \frac{x^{2} - 1}{x - 1} = \lim \frac{2 x}{1} \]
\[ \frac{2}{1} = 2 \]

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

x	f(x)	Distance from the limit
1.1	2.1000000	1.00e-1
0.9	1.9000000	1.00e-1
1.01	2.0100000	1.00e-2
0.99	1.9900000	1.00e-2
1.001	2.0010000	1.00e-3
0.999	1.9990000	1.00e-3
1.0001	2.0001000	1.00e-4
0.9999	1.9999000	1.00e-4
1.00001	2.0000100	1.00e-5
0.99999	1.9999900	1.00e-5

Questions

What is the limit of (

x^{2}

- 1) / (x - 1) as x approaches 1?

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

Which form is it?

0/0.

Limit of x2−1x−1\frac{x^{2} - 1}{x - 1}x−1x2−1​ as x→1x \to 1x→1

How to work it out

Values of (x2x^{2}x2 - 1) / (x - 1) as x → 1 (computed)

Questions

More well-known limits

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

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