Evaluate lim x → 1 (x / (x - 1))

To evaluate the limit as \( x \) approaches 1 for the expression \( \frac{x}{x - 1} \), we first substitute \( x = 1 \) directly into the function.

Direct Substitution

When we substitute, we get:

\( \frac{1}{1 - 1} = \frac{1}{0} \)

This results in an undefined expression, indicating that we need to analyze the limit further.

Approaching from the Left and Right

Next, we can examine the behavior of the function as \( x \) approaches 1 from both sides:

• From the left (x → 1^-): As \( x \) gets closer to 1 from values less than 1, \( x - 1 \) becomes a small negative number. Thus, \( \frac{x}{x - 1} \) approaches negative infinity.
• From the right (x → 1⁺): As \( x \) approaches 1 from values greater than 1, \( x - 1 \) becomes a small positive number. Therefore, \( \frac{x}{x - 1} \) approaches positive infinity.

Conclusion of the Limit

Since the left-hand limit approaches negative infinity and the right-hand limit approaches positive infinity, we conclude:

The limit does not exist.