What is the left-hand limit of 1/(x-1) as x approaches 1?

What is the left-hand limit of 1/(x-1) as x approaches 1?

• A. 0
• B. -∞
• C. ∞
• D. -1

Answer: (B)

Understanding what happens as x gets very close to 1 from values less than 1 is key. Here the denominator x−1 is a tiny negative number, so 1/(x−1) becomes a very large negative value. As the gap to 1 closes from the left, the expression dives toward negative infinity. So the left-hand limit is −∞. This contrasts with approaching from the right, which would give +∞, and explains why the function has a vertical asymptote at x = 1.

Understanding what happens as x gets very close to 1 from values less than 1 is key. Here the denominator x−1 is a tiny negative number, so 1/(x−1) becomes a very large negative value. As the gap to 1 closes from the left, the expression dives toward negative infinity. So the left-hand limit is −∞. This contrasts with approaching from the right, which would give +∞, and explains why the function has a vertical asymptote at x = 1.