find the limit.\n lim_{x\rightarrowinfty}\frac{sin 5x}{14x}\n lim_{x\rightarrowinfty}\frac{sin…

Explanation:

Step1: Recall sine - value range

The range of $\sin t$ is $[- 1,1]$, so $|\sin(5x)|\leq1$ for all $x$.

Step2: Analyze the limit

We have $\lim_{x\rightarrow\infty}\frac{\sin(5x)}{14x}$. We can rewrite it as $\frac{1}{14}\lim_{x\rightarrow\infty}\frac{\sin(5x)}{x}$. Since $|\sin(5x)|\leq1$, then $\left|\frac{\sin(5x)}{x}\right|\leq\frac{1}{x}$.

Step3: Apply the Squeeze Theorem

We know that $\lim_{x\rightarrow\infty}\frac{- 1}{x}=0$ and $\lim_{x\rightarrow\infty}\frac{1}{x}=0$. By the Squeeze Theorem, $\lim_{x\rightarrow\infty}\frac{\sin(5x)}{x}=0$.

Step4: Calculate the final limit

Since $\frac{1}{14}\lim_{x\rightarrow\infty}\frac{\sin(5x)}{x}$, and $\lim_{x\rightarrow\infty}\frac{\sin(5x)}{x}=0$, then $\lim_{x\rightarrow\infty}\frac{\sin(5x)}{14x}=0$.

Answer:

$0$