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  • Evaluate the limit <img alt="\lim _{x \rightarrow \infty}\left(\frac{\sin x \cos x}{x}\right)" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%20%5Cinfty%7D

Evaluate the limit \lim _{x \rightarrow \infty}\left(\frac{\sin x \cos x}{x}\right) with the help of sandwich theorem.

Option: 1

\pm \pi


Option: 2

1


Option: 3

0


Option: 4

\pi


Answers (1)

best_answer

The Sandwich theorem also known as squeeze play theorem applicable for any three real valued functions f(x),g(x) and h(x) that shares the same domain such that f(x)\leq g(x)\leq h(x) for \forall x in the domain of definition states the following: For some real value of  a, if \lim _{x \rightarrow a} f(x)=l=\lim _{x \rightarrow a} h(x) then \lim _{x \rightarrow a} f(x)=l 

Note that for  \forall x \in R, the range of the sine function is

\begin{aligned} & -1 \leq \sin (2 x) \leq 1 \\ & -1 \leq(2 \sin x \cos x) \leq 1 \\ & -\frac{1}{2} \leq(\sin x \cos x) \leq \frac{1}{2} \end{aligned}--------------(i)

Now, divide the inequality (i) by the variable that still preserves the inequality.

-\frac{1}{2 x} \leq\left(\frac{\sin x \cos x}{x}\right) \leq \frac{1}{2 x} -------------------(ii)

Apply the squeeze theorem to the inequality (ii).

\begin{aligned} & \lim _{x \rightarrow \infty}\left(-\frac{1}{2 x}\right) \leq \lim _{x \rightarrow \infty}\left(\frac{\sin x \cos x}{x}\right) \leq \lim _{x \rightarrow \infty}\left(\frac{1}{2 x}\right) \\ & 0 \leq \lim _{x \rightarrow \infty}\left(\frac{\sin x \cos x}{x}\right) \leq 0 \end{aligned}

Therefore, the required value of the limit is 

\lim _{x \rightarrow \infty}\left(\frac{\sin x \cos x}{x}\right)=0

 

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seema garhwal

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