Questions: Find the limit (if it exists). (If an answer does not exist, enter DNE.) lim x → 9- csc (πx/2)

Solution Steps

To find the limit of \(\csc \left(\frac{\pi x}{2}\right)\) as \(x\) approaches 9 from the left, we need to understand the behavior of the function near this point. The cosecant function, \(\csc(\theta) = \frac{1}{\sin(\theta)}\), becomes undefined when \(\sin(\theta) = 0\). We will evaluate the behavior of \(\sin\left(\frac{\pi x}{2}\right)\) as \(x\) approaches 9 from the left and determine if the limit exists.

To find the limit \(\lim_{x \to 9^-} \csc\left(\frac{\pi x}{2}\right)\), we first analyze the behavior of the function \(\csc(\theta) = \frac{1}{\sin(\theta)}\). The function becomes undefined when \(\sin(\theta) = 0\). Therefore, we need to evaluate \(\sin\left(\frac{\pi x}{2}\right)\) as \(x\) approaches 9 from the left.

Substitute \(x = 9\) into \(\frac{\pi x}{2}\) to get \(\frac{9\pi}{2}\). The sine function, \(\sin\left(\frac{9\pi}{2}\right)\), is equivalent to \(\sin\left(4\pi + \frac{\pi}{2}\right)\), which simplifies to \(\sin\left(\frac{\pi}{2}\right) = 1\).

Since \(\sin\left(\frac{\pi x}{2}\right)\) approaches 1 as \(x\) approaches 9 from the left, \(\csc\left(\frac{\pi x}{2}\right) = \frac{1}{\sin\left(\frac{\pi x}{2}\right)}\) approaches \(\frac{1}{1} = 1\).

Final Answer