Question

For each \[x\in \mathbb{R}\], let \[\left[ x \right]\] be the greatest integer less than or equal to \['x'\]. Then,
\[\displaystyle \lim_{x \to 0^{-}}\dfrac{x\left( \left[ x \right]+\left| x \right| \right)\sin \left[ x \right]}{\left| x \right|}\] is equal to
(a) \[-\sin 1\]
(b) 0
(c) 1
(d) \[\sin 1\]

Answer 1

Hint: We solve this problem first by replacing the internal functions with respective values then we find the limit of the function given. The function \[\left[ x \right]\] is called the step function defined as the greatest integer less than or equal to \['x'\]. Here, as \[x\to {{0}^{-}}\] then, \[\left[ x \right]\to -1\] because from the left of ‘0’ the greatest integer is ‘-1’. Also, the definition of modulus function is defined as
\[\left| x \right|=\left\{ \begin{align}
  & x,\forall x\ge 0 \\
 & -x,\forall x\le 0 \\
\end{align} \right.\]

Complete step by step answer:
We are asked to find the limit as
\[\displaystyle \lim_{x \to 0^{-}}\dfrac{x\left( \left[ x \right]+\left| x \right| \right)\sin \left[ x \right]}{\left| x \right|}\]
Let us assume the limit as
\[\Rightarrow L=\displaystyle \lim_{x \to 0^{-}}\dfrac{x\left( \left[ x \right]+\left| x \right| \right)\sin \left[ x \right]}{\left| x \right|}........equation(i)\]
We are given that\[\left[ x \right]\] be the greatest integer less than or equal to \['x'\] which is called the step function. We know that the step function always gives the greatest integer which is less than or equal to the value we need.
Here, we can see that in the limit \[x\to {{0}^{-}}\] which means the value of \['x'\] approaches to ‘0’ from the left side. So, we can say that the step function of \['x'\] is ‘-1’ because it is the only greatest integer less than zero.
\[\Rightarrow \left[ x \right]=-1\]
Now, we know that the modulus function is defined as
\[\left| x \right|=\left\{ \begin{align}
  & x,\forall x\ge 0 \\
 & -x,\forall x\le 0 \\
\end{align} \right.\]
Here, we know that the value of \['x'\] approaches to ‘0’ from left. So, we can say that
\[\Rightarrow \left| x \right|=-x\]
Now, by substituting the values of \[\left[ x \right],\left| x \right|\] in equation (i) we get
\[\Rightarrow L=\displaystyle \lim_{x \to 0^{-}}\dfrac{x\left( -1+\left( -x \right) \right)\sin \left( -1 \right)}{-x}\]
We know that \[\sin \left( -\theta \right)=-\sin \theta \]. By using this result in above equation we get
\[\begin{align}
  & \Rightarrow L=\displaystyle \lim_{x \to 0^{-}}\dfrac{-x\left( 1+x \right)\left( -\sin 1 \right)}{-x} \\
 & \Rightarrow L=\left( -\sin 1 \right)\displaystyle \lim_{x \to 0^{-}}\left( 1+x \right) \\
 & \Rightarrow L=\left( -\sin 1 \right)\left( 1+0 \right) \\
 & \Rightarrow L=-\sin 1 \\
\end{align}\]
Therefore the limit of given function is given as
\[\displaystyle \lim_{x \to 0^{-}}\dfrac{x\left( \left[ x \right]+\left| x \right| \right)\sin \left[ x \right]}{\left| x \right|}=-\sin 1\]

So, the correct answer is “Option a”.

Note: Students will make mistakes in taking the value of step function. Here the value of \['x'\] approaches to ‘0’ from left, so that the value of \[\left[ x \right]\] will be ‘-1’. Since the definition of step function says that the value of \[\left[ x \right]\] is greatest integer less than or equal to \['x'\], students may take the value of \[\left[ x \right]\] as ‘0’. This gives the wrong answer because while \['x'\] approaches to 0 it is not necessary that the value of \['x'\] will be ‘0’. So, we cannot consider that as the correct answer. This part needs to be taken care of.