Evaluate the Limit limit as x approaches 1 of 1/(x-1)

Problem

(lim_x→1)(1/(x−1))

Solution

1. Analyze the behavior of the function as x approaches 1 from the right (x→1. As x becomes slightly larger than 1 the denominator x−1 is a very small positive number, causing the fraction to increase without bound.

(lim_x→1)(1/(x−1))=∞

2. Analyze the behavior of the function as x approaches 1 from the left (x→1. As x becomes slightly smaller than 1 the denominator x−1 is a very small negative number, causing the fraction to decrease without bound.

(lim_x→1)(1/(x−1))=−∞

3. Compare the one-sided limits. Since the limit from the right and the limit from the left do not approach the same value, the two-sided limit does not exist.

∞≠−∞

Final Answer

(lim_x→1)(1/(x−1))=Does Not Exist

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