Question

The value of $\displaystyle \lim_{x \to 0}$ $\frac{\left(1-cos\,2x\right)sin\,5x}{x^{2}\,sin\,3x}$ is

• A. $\frac{10}{3}$
• B. $\frac{3}{10}$
• C. $\frac{6}{5}$
• D. $\frac{5}{6}$

Answer 1

Answer: (A)

$\displaystyle \lim_{x \to 0}$ $\frac{\left(1-cos\,2x\right)sin\,5x}{x^{2}\,sin\,3x}$
$=\displaystyle \lim_{x \to 0}$ $\frac{2\,sin^{2}\,x\,sin\,5x}{x^{2}\,sin\,3x}$
$=\displaystyle \lim_{x \to 0}$$\left(\frac{2\,sin^{2}\,x}{x^{2}}\right) \frac{\left(\frac{sin\,5x}{5x}\right)\times5}{\left(\frac{sin\,3x}{3x}\right)\times3}$
$=\displaystyle \lim_{x \to 0}$ $\left(\frac{sin\,x}{x}\right)^2\times \:\frac{5\displaystyle \lim_{5x \to 0} \left(\frac{sin\,5x}{5x}\right)}{3 \displaystyle \lim_{3x \to 0} \left(\frac{sin\,3x}{3x}\right)}$
$=\frac{2 \times 5}{3}=\frac{10}{3}$