# Evaluate the limit #lim_(x→0)(sin x/x)^(1/x^2)#?

correspondingly

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To evaluate the limit ( \lim_{x \to 0} \left( \frac{\sin x}{x} \right)^{\frac{1}{x^2}} ), follow these steps:

- Rewrite the limit as ( \lim_{x \to 0} \left( \left( \frac{\sin x}{x} \right)^{\frac{x^2}{\sin x}} \right)^{\frac{\sin x}{x} \cdot \frac{1}{x^2}} ).
- Notice that ( \lim_{x \to 0} \frac{\sin x}{x} = 1 ), so the expression becomes ( \left( 1^{\frac{x^2}{\sin x}} \right)^{\frac{1}{x^2}} ).
- Simplify further to ( \lim_{x \to 0} 1^{\frac{1}{x^2}} ).
- Since ( 1^{\text{anything}} ) is always 1, the limit is ( \boxed{1} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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