Evaluate the Limit limit as x approaches 3 of tan((pix)/4)

Problem

(lim_x→3)(tan((π*x)/4))

Solution

1. Identify the type of limit by checking if the function is continuous at the target value x=3

2. Substitute the value x=3 directly into the expression, as the tangent function is defined for this input.

tan((π*(3))/4)

3. Simplify the argument of the tangent function.

tan((3*π)/4)

4. Evaluate the trigonometric value using the unit circle, noting that (3*π)/4 is in the second quadrant where tangent is negative.

tan((3*π)/4)=−1

Final Answer

(lim_x→3)(tan((π*x)/4))=−1

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