# How do you find #lim sin(2x)/x# as #x->0# using l'Hospital's Rule?

The limit:

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To find ( \lim_{{x \to 0}} \frac{{\sin(2x)}}{x} ) using l'Hopital's Rule, we can differentiate the numerator and denominator separately with respect to ( x ) until we can evaluate the limit directly:

( \lim_{{x \to 0}} \frac{{\sin(2x)}}{x} )

Applying l'Hopital's Rule once:

( = \lim_{{x \to 0}} \frac{{2\cos(2x)}}{1} )

Evaluating at ( x = 0 ), we get:

( = 2\cos(0) = 2 )

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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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