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  • Find this limit with the help of the squeeze theorem. <img alt="\lim _{x \rightarrow 0}(x \cos x), \quad \forall x \in R" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7

Find this limit with the help of the squeeze theorem.

\lim _{x \rightarrow 0}(x \cos x), \quad \forall x \in R

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(\mathrm{x}) \leq g(\mathrm{x}) \leq h(\mathrm{x}) for \forall \mathrm{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} g(x)=l

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

-1 \leq \cos x \leq 1  ______(1)

Now, multiply the inequality (i) by the absolute value of that still preserves the inequality.

\begin{aligned} & |x| \times(-1) \leq|x| \times \cos x \leq|x| \times 1 \\ & -|x| \leq|x| \cos x \leq|x| \end{aligned}------------(2)

Apply the squeeze theorem to the inequality (ii).

\begin{aligned} & \lim _{x \rightarrow 0}(-|x|) \leq \lim _{x \rightarrow 0}(x \cos x) \leq \lim _{x \rightarrow 0}|x| \\ & 0 \leq \lim _{x \rightarrow 0}(x \cos x) \leq 0 \end{aligned}

Therefore, the required value of the limit is 

\lim _{x \rightarrow 0}(x \cos x)

 

 

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