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, $$ y = 	an x $$, near $$ x = \frac{\pi}{2} . $$The tangent function has vertical asymptotes at $$ x = \frac{\pi}{2} + k\pi $$ for any integer $$ k $$, where the function approaches infinity as $$ x $$ approaches these points from the left and negative infinity from the right. Step 2: Consider the limit $$ \lim_{x 	o \frac{\pi}{2}^-} 	an x . $$This notation $$ x 	o \frac{\pi}{2}^- $$ indicates that $$ x  is $$approaching $$ \frac{\pi}{2} $$ from the left side, meaning $$ x  is $$slightly less than $$ \frac{\pi}{2} . $$Step 3: Analyze the behavior of $$ 	an x  as$$ $$ x $$ approaches $$ \frac{\pi}{2} $$ from the left. As $$ x $$ gets closer to $$ \frac{\pi}{2} $$, the value of $$ 	an x $$ increases without bound, indicating that the limit approaches 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 $$ x = \frac{\pi}{2} $$, where the function approaches infinity from the left and negative infinity from the right. Step 5: Use the graph to verify your analysis. Observe that as $$ x $$ approaches $$ \frac{\pi}{2} $$ from the left, the graph of $$ 	an x $$ indeed rises steeply towards positive infinity, confirming the limit $$ \lim_{x 	o \frac{\pi}{2}^-} 	an x = +\infty .$$

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

Vertical Asymptotes

Graphing Trigonometric Functions