Question

# Consider the given function, $f(x)=\left[ \begin{matrix} \dfrac{{{\tan }^{2}}\left\{ x \right\}}{{{x}^{2}}-{{\left[ x \right]}^{2}}} \\ 1 \\ \sqrt{\left\{ x \right\}\cot \left\{ x \right\}} \\\end{matrix}\begin{matrix} for\text{ }x>0\text{ } \\ for\text{ }x=0\text{ } \\ for\text{ }x<0\text{ } \\\end{matrix} \right.$where$\left[ x \right]$is the step up function and$\left\{ x \right\}$ is the fractional part function of$x$, then:(a) $\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f(x)=1$(b)$\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=1$(c) ${{\cot }^{-1}}{{\left( \underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x) \right)}^{2}}=1$(d) $f\text{ is continuous at }x=1$

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Hint: Apply limit to the given function separately at point x = 0 and x = 1 and then substitute $\{x\}+\left[ x \right]=x$, $\underset{x\to 0}{\mathop{\lim }}\,\dfrac{\tan x}{x}=1$ and $\underset{x\to 0}{\mathop{\lim }}\,\dfrac{\sin x}{x}=1$, simplify it further. Then check the validity of the options by using various properties of the limit.

We are given the function $f(x)=\left[ \begin{matrix} \dfrac{{{\tan }^{2}}\left\{ x \right\}}{{{x}^{2}}-{{\left[ x \right]}^{2}}} \\ 1 \\ \sqrt{\left\{ x \right\}\cot \left\{ x \right\}} \\ \end{matrix}\begin{matrix} for\text{ }x>0\text{ } \\ for\text{ }x=0\text{ } \\ for\text{ }x<0\text{ } \\ \end{matrix} \right.$
We will apply the limit to the given function around the point $x=0$ under various conditions and then check the continuity of the function around the point $x=1$.
We know that $\left\{ x \right\}$ is the function that evaluates the fractional value of $x$ and $\left[ x \right]$ is the function that evaluates the integral value of $x$.
For$x>0$, we have the function$f(x)$such that$f\left( x \right)=\dfrac{{{\tan }^{2}}\left\{ x \right\}}{{{x}^{2}}-{{\left[ x \right]}^{2}}}$.
Thus, by applying the limit on the given function, we get
$\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\dfrac{{{\tan }^{2}}\left\{ x \right\}}{{{x}^{2}}-{{\left[ x \right]}^{2}}}$
Now we will apply left hand limit using the formula, $f\left( {{0}^{+}} \right)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{f(0+h)-f(0)}{0+h}$, we get
\begin{align} & \underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f(x)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{{{\tan }^{2}}\left\{ 0+h \right\}}{{{\left( 0+h \right)}^{2}}-{{\left[ 0+h \right]}^{2}}} \\ & \Rightarrow \underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f(x)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{{{\tan }^{2}}h}{{{h}^{2}}} \\ \end{align}
As, we know that $\underset{x\to 0}{\mathop{\lim }}\,\dfrac{\tan x}{x}=1$, so, we get $\underset{x\to 0}{\mathop{\lim }}\,\dfrac{{{\tan }^{2}}x}{{{x}^{2}}}=1$ as well.
Thus, we get $\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\dfrac{{{\tan }^{2}}\left\{ x \right\}}{{{\left\{ x \right\}}^{2}}}=1$.
Hence, we have the value of limit as $\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\dfrac{{{\tan }^{2}}\left\{ x \right\}}{{{x}^{2}}-{{\left[ x \right]}^{2}}}=1$.
Now, we will consider the case $x<0$. For $x<0$, we have the function $f(x)$ such that $f\left( x \right)=\sqrt{\left\{ x \right\}\cot \left\{ x \right\}}$.
Applying the limit on the given function, we get
$\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\sqrt{\left\{ x \right\}\cot \left\{ x \right\}}$
Further simplifying the limit, we have
$\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\sqrt{\left\{ x \right\}\dfrac{\cos \left\{ x \right\}}{\sin \left\{ x \right\}}}$ as we know that $\cot x=\dfrac{\cos x}{\sin x}$.
As we know that $\underset{x\to 0}{\mathop{\lim }}\,\dfrac{\sin x}{x}=1$, we have$\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\sqrt{\left\{ x \right\}\dfrac{\cos \left\{ x \right\}}{\sin \left\{ x \right\}}}=\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\sqrt{\cos \left\{ x \right\}}$
Thus, we have
$\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\sqrt{\cos \left\{ x \right\}}=\sqrt{\cos 0}=1$
Hence, we have
$\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\sqrt{\left\{ x \right\}\cot \left\{ x \right\}}=1$
Now, we need to evaluate the value of
${{\cot }^{-1}}{{\left( \underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x) \right)}^{2}}$
As $\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\sqrt{\left\{ x \right\}\cot \left\{ x \right\}}=1$, we have
${{\cot }^{-1}}{{\left( \underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x) \right)}^{2}}={{\cot }^{-1}}{{\left( 1 \right)}^{2}}={{\cot }^{-1}}1=\dfrac{\pi }{4}$
Now, we will check the continuity of $f$ at point $x=1$ as for $x=1$, we have
$f\left( {{x}^{-}} \right)=f\left( {{x}^{+}} \right)$.
Thus, the function$f$is continuous at $x=1$

So, the correct answers are “Option A, B and D”.

Note: It’s necessary to evaluate both left- and right-hand side of the limit around a point. Otherwise, we won’t get a correct answer if we apply only one side of the limit.
Students sometimes substitute $\{x\}+\left[ x \right]=x\Rightarrow \left[ x \right]=x-\{x\}$, in this way the process will get lengthy and chances of getting the wrong solution is there.