Evaluate the Limit limit as x approaches 5 of cos((pix)/6)

Problem

(lim_x→5)(cos((π*x)/6))

Solution

1. Identify the type of function. The function ƒ(x)=cos((π*x)/6) is a composition of a trigonometric 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=5 the limit is equal to the function value at that point.

3. Substitute the value x=5 into the expression.

cos((π⋅5)/6)

4. Simplify the argument of the cosine function.

cos((5*π)/6)

5. Evaluate the trigonometric value. The angle (5*π)/6 is in the second quadrant, where the cosine value is negative.

cos((5*π)/6)=−√(,3)/2

Final Answer

(lim_x→5)(cos((π*x)/6))=−√(,3)/2

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