Using radians, determine the value approached by sin(5x)/(2x) when x tends to 0. Write it as a reduced fraction.

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Sine of a small angle

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Using radians, determine the value approached by $\frac{sin ( 5 x )}{2 x}$ when $x$ tends to $0$ . Write it as a reduced fraction.

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Hint

Rewrite the ratio so that the standard limit of $\frac{sin u}{u}$ appears.

Solution

Write $\frac{sin ( 5 x )}{2 x} = (\frac{5}{2}) \times \frac{sin ( 5 x )}{5 x}$ .
As $x \to 0$ , also $5 x \to 0$ , and $\frac{sin u}{u} \to 1$ as $u \to 0$ .
So the limit is $\frac{5}{2} \times 1 = \frac{5}{2}$ .

Answer: $\frac{5}{2}$