Questions: Find the limit of the transcendental function. lim x→1 cos(πx/3)

Transcript text: Find the limit of the transcendental function. \[ \lim _{x \rightarrow 1} \cos \left(\frac{\pi x}{3}\right) \]

Solution

Solution Steps

To find the limit of the given transcendental function as \( x \) approaches 1, we can directly substitute \( x = 1 \) into the function, since cosine is continuous everywhere. This will give us the value of the limit.

Step 1: Evaluate the Function at the Limit Point

To find the limit of the function \( \lim_{x \rightarrow 1} \cos \left(\frac{\pi x}{3}\right) \), we substitute \( x = 1 \) into the function. This is possible because the cosine function is continuous everywhere.

Step 2: Substitute and Simplify

Substitute \( x = 1 \) into the function: \[ \cos \left(\frac{\pi \times 1}{3}\right) = \cos \left(\frac{\pi}{3}\right) \]

Step 3: Calculate the Cosine Value

The value of \( \cos \left(\frac{\pi}{3}\right) \) is known to be \( \frac{1}{2} \).