Evaluate the Limit limit as x approaches 0 of e^(1/x)

Problem

(lim_x→0)(e1/x)

Solution

1. Identify the behavior of the exponent as x approaches 0 from different sides, as the limit depends on whether x is positive or negative.

2. Evaluate the right-hand limit by letting x→0 As x approaches 0 from the right, the exponent 1/x approaches ∞

(lim_x→0)(1/x)=∞

3. Determine the limit of the exponential function for the right-hand side.

(lim_x→0)(e1/x)=e∞=∞

4. Evaluate the left-hand limit by letting x→0 As x approaches 0 from the left, the exponent 1/x approaches −∞

(lim_x→0)(1/x)=−∞

5. Determine the limit of the exponential function for the left-hand side.

(lim_x→0)(e1/x)=e(−∞)=0

6. Compare the one-sided limits. Since the right-hand limit (∞ and the left-hand limit (0 are not equal, the general limit does not exist.

Final Answer

(lim_x→0)(e1/x)=Does Not Exist

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