Question

evaluate this question​

Answer 1

Answer:

$\boxed{\lim_{x\rightarrow 0} \dfrac{\text{sin}(x^{2})}{x(\text{sin}x)}=1}$

Step-by-step explanation:

The given expression is as follows:

$\boxed{\lim_{x\rightarrow 0} \dfrac{\text{sin}(x^{2})}{x(\text{sin}x)}}$

The property used to solve the above expression is as shown below:

$\boxed{\lim_{x\rightarrow0}\dfrac{\text{sin}(x)}{x}=1}$

Solve the given expression as shown below:

$\boxed{\begin{aligned}\lim_{x\rightarrow 0} \dfrac{\text{sin}(x^{2})}{x(\text{sin}x)}&=\lim_{x\rightarrow0}\left(\dfrac{\text{sin}(x)^{2}}{x\cdot x\cdot \frac{1}{x}\cdot \text{sin}x}\right)\\&=\lim_{x\rightarrow0}\left(\dfrac{\text{sin}(x)^{2}}{x^{2}}\right)\cdot \lim_{x\rightarrow0}\left(\dfrac{1}{\frac{\text{sin}x}{x}}\right)\\&=1\cdot 1\\&=1\end{aligned}}$

Therefore, the simplified value of the given expression is $\boxed{\bf 1}$.