The function $$f(x)=xtan$$⁡$$2xsin$$⁡$$3x$$⋅$$sin$$⁡$$5x$$ for x≠$$0$$ is continuous at $$x=0$$ then $$f(0)$$

$$f0=K=Ltx$$→$$0$$ $$xtan2xsin3x$$.$$sin5x=Ltx$$→$$0$$ $$xtan2xx2sin3xx$$.$$sin5xx=215$$

Continuity

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Question

$The function f (x) = \frac{x \tan 2 x}{\sin 3 x \cdot \sin 5 x} for x \neq 0 is continuous at x = 0 then f (0)$

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Solution

$f (0) = K = \underset{x \to 0}{L t} \frac{x \tan 2 x}{\sin 3 x . \sin 5 x}$
$\begin{array}{l} = \underset{x \to 0}{L t} \frac{\frac{x \tan 2 x}{x^{2}}}{\frac{\sin 3 x}{x} . \frac{\sin 5 x}{x}} \\ = \frac{2}{15} \end{array}$

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