# How do you find the limit #lim_(x->0)sin(x)/x# ?

We will use l'Hôpital's Rule.

l'Hôpital's rule states:

Thus,

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To find the limit of ( \lim_{x \to 0} \frac{\sin(x)}{x} ), you can use L'Hôpital's Rule or the Maclaurin series expansion of ( \sin(x) ). Applying L'Hôpital's Rule yields ( \lim_{x \to 0} \frac{\cos(x)}{1} ), which evaluates to ( \cos(0) = 1 ). Alternatively, you can use the Maclaurin series expansion of ( \sin(x) ) to get ( \lim_{x \to 0} \frac{x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots}{x} ), simplifying to ( \lim_{x \to 0} \left(1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \ldots\right) ), which equals ( 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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