Evaluate the Limit limit as x approaches pi/2 of tan(x)

Problem

(lim_x→π/2)(tan(x))

Solution

1. Identify the function and the point of approach. The function is ƒ(x)=tan(x) and we are approaching x=π/2

2. Express the tangent function in terms of sine and cosine to analyze the behavior near the limit point.

tan(x)=sin(x)/cos(x)

3. Evaluate the limits of the numerator and denominator separately as x approaches π/2

(lim_x→π/2)(sin(x))=1

(lim_x→π/2)(cos(x))=0

4. Analyze the behavior from both sides. As x approaches π/2 from the left (x→π/2−, cos(x) is positive and approaching 0 so the ratio approaches +∞

(lim_x→π/2−)(tan(x))=∞

5. Analyze the behavior from the right (x→π/2+. As x approaches π/2 from the right, cos(x) is negative and approaching 0 so the ratio approaches −∞

(lim_x→π/2+)(tan(x))=−∞

6. Conclude that since the left-hand limit and the right-hand limit are not equal, the two-sided limit does not exist.

Final Answer

(lim_x→π/2)(tan(x))=Does Not Exist

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