Evaluate the Limit limit as x approaches 5 of 1/(x-5)

Problem

(lim_x→5)(1/(x−5))

Solution

1. Analyze the behavior of the function as x approaches 5 from the right. As x becomes slightly larger than 5 the denominator x−5 is a very small positive number, which causes the fraction to increase without bound.

(lim_x→5)(1/(x−5))=∞

2. Analyze the behavior of the function as x approaches 5 from the left. As x becomes slightly smaller than 5 the denominator x−5 is a very small negative number, which causes the fraction to decrease without bound.

(lim_x→5)(1/(x−5))=−∞

3. Compare the one-sided limits. Since the limit from the right and the limit from the left are not equal, the two-sided limit does not exist.

∞≠−∞

Final Answer

(lim_x→5)(1/(x−5))=Does not exist

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