Questions: The graph of the function f(x)=tan x is given above for the interval x ∈[0,2 π] ONLY. Determine the one-sided limit. Then indicate the equation of the vertical asymptote. Find lim x→(π/2)− f(x)= □ This indicates the equation of a vertical asymptote is x= □ Find lim x→(3π/2)+ f(x)= □ This indicates the equation of a vertical asymptote is x= □

Solution Steps

The function given is \( f(x) = \tan x \) and the interval is \( x \in [0, 2\pi] \).

The tangent function has vertical asymptotes where the cosine function is zero, which occurs at \( x = \frac{\pi}{2} \) within the given interval. We need to find the limit as \( x \) approaches \( \frac{\pi}{2} \) from the left: \[ \lim_{x \to \left(\frac{\pi}{2}\right)^-} \tan x = +\infty \]

Since the limit approaches \( +\infty \), there is a vertical asymptote at \( x = \frac{\pi}{2} \).

Similarly, the tangent function has another vertical asymptote at \( x = \frac{3\pi}{2} \). We need to find the limit as \( x \) approaches \( \frac{3\pi}{2} \) from the left: \[ \lim_{x \to \left(\frac{3\pi}{2}\right)^-} \tan x = +\infty \]

Since the limit approaches \( +\infty \), there is a vertical asymptote at \( x = \frac{3\pi}{2} \).

Final Answer