Evaluate the Limit limit as x approaches 3 of arctan(e^x)

Problem

(lim_x→3)(arctan(ex))

Solution

1. Identify the type of function. The expression arctan(ex) is a composition of two continuous functions: the exponential function ex and the inverse tangent function arctan(u)

2. Apply the property of limits for continuous functions. Since both functions are continuous everywhere on their domains, the limit can be evaluated by direct substitution.

3. Substitute the value x=3 into the expression.

L=arctan(e3)

Final Answer

(lim_x→3)(arctan(ex))=arctan(e3)

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