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Answer :
Final answer:
The original problem is the limit of (sinx)^x as x approaches 0. By rewriting the formula and applying the rules of limits, we deduced that the limit is equal to 1/e, or approximately 0.368.
Explanation:
The question is about finding the limit of (sinx)^x as x tends to 0 from the positive side. In mathematics, this is a classic limit problem often encountered in calculus. It uses the limit laws, the concept of continuity, and L'Hopital's rule to solve.
To solve this problem, we'll use a well-known limit identity, which is: lim (x→0 ) (1 + a*x)^(1/x) = e^a
To apply this rule, we need to rewrite our expression in the form of this identity. Let a = sinx - 1 and x = sinx. Our limit function becomes:
lim (x→0 ) (1 + (sinx - 1))^(1/sinx)
This expression now falls into the limit identity form and hence, as x approaches 0 from the positive side, this limit equals e^(sinx - 1).
As x → 0, sinx - 1 → -1. Therefore, the value of the limit is e^(-1), or 1/e, which is approximately 0.368.
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