Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Find this limit with the help of the squeeze theorem. <img alt="\lim _{x \rightarrow 0}\left[x^2 \sin (\pi x)\right], \quad \forall x \in R" src="https://entrancecorne

Find this limit with the help of the squeeze theorem.

\lim _{x \rightarrow 0}\left[x^2 \sin (\pi x)\right], \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 \nabla x in the domain of definition states the following:

For some real value of \alpha 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

-1 \leq \sin (\pi x) \leq 1 ---------(1)

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

\begin{aligned} & \left(x^2\right) \times(-1) \leq\left(x^2\right) \times \sin (\pi x) \leq\left(x^2\right) \times 1 \\ & -x^2 \leq x^2 \sin (\pi x) \leq x^2 \\ & -x^2 \leq x^2 \sin (\pi x) \leq x^2 \quad \ldots(i i) \end{aligned}

Apply the squeeze theorem to the inequality (ii).

\begin{aligned} & \lim _{x \rightarrow 0}\left(-x^2\right) \leq \lim _{x \rightarrow 0}\left(x^2 \sin (\pi x)\right) \leq \lim _{x \rightarrow 0} x^2 \\ & \left(\lim _{x \rightarrow 0}(-x)\right)^2 \leq \lim _{x \rightarrow 0}\left(x^2 \sin (\pi x)\right) \leq\left(\lim _{x \rightarrow 0} x\right)^2 \\ & 0 \leq \lim _{x \rightarrow 0}\left(x^2 \sin (\pi x)\right) \leq 0 \end{aligned}

Therefore, the required value of the limit is

\lim _{x \rightarrow 0}\left(x^2 \sin (\pi x)\right)=0

 

 

 

Posted by

Ritika Jonwal

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2025: BTech in AI gains popularity; IIT, NIT cut-offs for artificial intelligence
anam-khan

JEE Main 2025 Live: NTA JEE session 1 begins tomorrow; admit card link, exam pattern, passing marks
anam-khan

JEE Main 2025 exam centre changed at UAE's Sharjah due to “unavoidable circumstances”

Latest Question