Evaluate the Limit limit as x approaches 1 of sin(pix)

Problem

(lim_x→1)(sin(π*x))

Solution

1. Identify the type of function. The function ƒ(x)=sin(π*x) 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. Since the function is continuous at x=1 the limit is equal to the value of the function at that point.

3. Substitute x=1 into the expression.

sin(π⋅1)=sin(π)

4. Evaluate the trigonometric value. The sine of π radians (or 180 degrees) is 0.

sin(π)=0

Final Answer

(lim_x→1)(sin(π*x))=0

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