Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.lim x→π/2^− tan x

Accepted Answer

Step 1: Understand the behavior of the tangent function. The function $$ y = 	an x $$ has vertical asymptotes where the cosine of $$ x  is $$zero, which occurs at $$ x = \frac{\pi}{2} + k\pi $$ for any integer $$ k . $$Step 2: Identify the limit direction. The limit $$ \lim_{x 	o \frac{\pi}{2}^-} 	an x $$ means we are approaching $$ \frac{\pi}{2} $$ from the left side, or from values less than $$ \frac{\pi}{2} . $$Step 3: Analyze the behavior of $$ 	an x  as$$ $$ x $$ approaches $$ \frac{\pi}{2} $$ from the left. As $$ x $$ approaches $$ \frac{\pi}{2}^- $$, $$ 	an x $$ increases without bound because the cosine of $$ x $$ approaches zero from the positive side, making $$ 	an x = \frac{\sin x}{\cos x} $$ approach positive infinity. Step 4: Sketch the graph of $$ y = 	an x $$ over the interval $$[-\pi, \pi]. $$Note the vertical asymptotes at $$ x = -\frac{\pi}{2} $$ and $$ x = \frac{\pi}{2} $$, and the periodic nature of the tangent function with period $$ \pi . $$Step 5: Use the graph to verify the limit. On the graph, observe that as $$ x $$ approaches $$ \frac{\pi}{2} $$ from the left, the value of $$ 	an x $$ increases towards positive infinity, confirming the limit analysis.

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.
lim x→π/2^− tan x

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Key Concepts

Limits

Tangent Function

Graphing Functions