Evaluate the Limit limit as x approaches 1 of cos((pix)/3)

Problem

(lim_x→1)(cos((π*x)/3))

Solution

1. Identify the type of function. The function ƒ(x)=cos((π*x)/3) is a composition of a cosine function and a linear function, both of which are continuous everywhere.

2. Apply the direct substitution property for limits of continuous functions. Since the function is continuous at x=1 the limit is equal to the function value at that point.

3. Substitute the value x=1 into the expression.

cos((π*(1))/3)

4. Simplify the argument of the cosine function.

cos(π/3)

5. Evaluate the trigonometric value using the unit circle or special right triangles. The cosine of π/3 radians (or 60 is 1/2

Final Answer

(lim_x→1)(cos((π*x)/3))=1/2

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