# How do you find #lim_(theta->0) tantheta/theta# using l'Hospital's Rule?

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Determine that the expression evaluated at the limit results in an indeterminate form.

Differentiate the number and denominator.

The limit of the new expression is the same as the original.

Therefore, the limit of the original expression is same:

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To find (\lim_{\theta \to 0} \frac{\tan(\theta)}{\theta}) using l'Hospital's Rule, we first rewrite the expression as (\frac{\sin(\theta)}{\theta \cdot \cos(\theta)}). Then, we differentiate the numerator and denominator separately with respect to (\theta). After applying l'Hospital's Rule, we get (\lim_{\theta \to 0} \frac{\sin'(\theta)}{\theta \cdot \cos'(\theta)} = \lim_{\theta \to 0} \frac{\cos(\theta)}{\cos(\theta) - \theta \cdot (-\sin(\theta))}). Simplifying further, this yields (\lim_{\theta \to 0} \frac{\cos(\theta)}{\cos(\theta) + \theta \sin(\theta)}). As (\theta) approaches 0, both the numerator and denominator approach 1, thus the limit is 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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