Final Answer:
The one-sided limit as x approaches 4 from the positive direction of sec(πx/8) is infinity.
Explanation:
As x approaches 4 from the positive side, the angle within sec(πx/8) approaches π. The secant function, which is the reciprocal of the cosine function, has a vertical asymptote at odd multiples of π/2. Therefore, as the input angle of secant reaches π, the function approaches infinity. This occurs due to the cosine function approaching zero at π, resulting in the reciprocal becoming extremely large, hence the limit is infinity. The nature of secant's asymptotic behavior around π and the reciprocal relationship with cosine confirms the unbounded growth of the function as x tends towards 4 from the positive side.
The behavior of secant function around the π/2, 3π/2, etc., multiples causes it to tend toward positive or negative infinity. In this specific case, as x approaches 4 from the positive direction, the angle within sec(πx/8) approaches π, thus leading to an unbounded increase, indicating that the one-sided limit is infinity.