Let be defined for

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let f(x)=tan⁡xf(x)=\tan x be defined for −π2<x<π2-\tfrac{\pi}{2}<x<\tfrac{\pi}{2}. The function has a vertical asymptote at x=π2x=\tfrac{\pi}{2}. (a) Determine lim⁡x→(π2)−f′(x)\displaystyle \lim_{x\to\left(\frac{\pi}{2}\right)^{-}}f'(x). Provide a clear justification based on the behavior of the slope of the tangent line as xx approaches π2\tfrac{\pi}{2} from the left. (b) Describe, in complete sentences, how the result from part (a) is reflected in the geometric appearance of the graph of ff near x=π2x=\tfrac{\pi}{2}. Comment specifically on the orientation of tangent lines drawn at points with xx–coordinates near (but less than) π2\tfrac{\pi}{2}. No sketches are required, but any verbal description should be precise.

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