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Evaluate the given limit

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Controldrag $\underset{x\to 0}{lim}\dots$
Context Panel: Evaluate and Display Inline


$\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(x\right)}{x}$ = ${1}$${}$

Apply L'Hôpital's rule

$\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(x\right)}{x}\=\underset{x\to 0}{lim}\frac{\mathrm{cos}\left(x\right)}{1}\=\underset{x\to 0}{lim}\mathrm{cos}\left(x\right)\=1$



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The application of L'Hôpital's rule is valid because as $x\to 0$, the fraction $\mathrm{sin}\left(x\right)\/x$ tends to the indeterminate form $0\/0$.
This limit is easily obtained by L'Hôpital's rule. However, to apply this rule, the derivative of $\mathrm{sin}\left(x\right)$ must already be known. The derivation of this derivative is given in Table 2.7.5, which in turn invokes the special trig limits in Table 1.4.1. On a logical basis, these earlier calculations were essential, and could not have been deleted in favor of L'Hôpital's rule without tearing a hole in the logical structure of the calculus.
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