# The function $f\left( x \right)=x{{\tan }^{-1}}\dfrac{1}{x}$ for $x\ne 0$ , $f\left( 0 \right)=0$ is

A. Differentiable at x = 0

B. neither continuous at x = 0 nor differentiable at x = 0

C. Not continuous at x = 0

D. continuous at x = 0 but not differentiable at x = 0

Answer

Verified

278.4k+ views

**Hint:**We need to find whether the function $f\left( x \right)=x{{\tan }^{-1}}\dfrac{1}{x}$ is differentiable and continuous at x = 0. We solve the given question by finding out the range of the given function and apply limits to get the desired result.

**Complete step-by-step solution:**

We are given a function $f\left( x \right)=x{{\tan }^{-1}}\dfrac{1}{x}$ and are asked to find if the function is differentiable and continuous at x = 0. We will be solving the given question using the concept of limits and

differentiation.

We will have to find the range of the given function $f\left( x \right)=x{{\tan }^{-1}}\dfrac{1}{x}$ .

From the formulae of trigonometry, we know that the range of the function ${{\tan }^{-1}}\dfrac{1}{x}$ is $\left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right)$

Writing the same in the form of inequality, we get,

$\Rightarrow -\dfrac{\pi }{2} < {{\tan }^{-1}}\dfrac{1}{x} < \dfrac{\pi }{2}$

Multiplying the inequality with x, we get,

$\Rightarrow -\dfrac{\pi }{2}x < x{{\tan }^{-1}}\dfrac{1}{x} < x\dfrac{\pi }{2}$

A function $f\left( x \right)$ is continuous if, for every value of c in the domain, the function $f\left( c \right)$ is defined and

$\Rightarrow \displaystyle \lim_{x \to c}f\left( x \right)=f\left( c \right)$

We need to find the limit of the function $f\left( x \right)=x{{\tan }^{-1}}\dfrac{1}{x}$ .

Following the same, we get,

$\Rightarrow \displaystyle \lim_{x \to 0}f\left( x \right)=\displaystyle \lim_{x \to 0}x{{\tan }^{-1}}\dfrac{1}{x}$

Applying the limits to the above equation, we get,

$\Rightarrow \displaystyle \lim_{x \to 0}f\left( x \right)=\displaystyle \lim_{x \to 0}x{{\tan }^{-1}}\dfrac{1}{x}=0$

From the above,

$\therefore \displaystyle \lim_{x \to 0}x{{\tan }^{-1}}\dfrac{1}{x}=0$

And from the question we know that $f\left( 0 \right)=0$

$\therefore f\left( x \right)$ is continuous at x = 0.

We can test if the function is differentiable at any point c using the limit given as follows,

$\Rightarrow \displaystyle \lim_{h\to 0}\dfrac{f\left( c+h \right)-f\left( c \right)}{h}$

If the limit exists, the function is differentiable.

As per the given question, we get,

$\Rightarrow \displaystyle \lim_{x \to 0}\dfrac{f\left( x \right)-f\left( 0 \right)}{x-0}$

Substituting the values in the above expression, we get,

$\Rightarrow \displaystyle \lim_{x \to 0}\dfrac{x{{\tan }^{-1}}\dfrac{1}{x}-0}{x}$

Simplifying the above equation, we get,

$\Rightarrow \displaystyle \lim_{x \to 0}{{\tan }^{-1}}\dfrac{1}{x}$

We know that the value of the limit ${{\tan }^{-1}}\dfrac{1}{x}$ does not exist at $x=0$ . So, the function is not differentiable at x = 0.

**$\therefore$ Option D holds the correct answer for the given question.**

**Note:**We must know the basic formulae of derivatives and limits to solve the given question easily.

The continuity of the function given in the question can also be determined by plotting the graph of

the function. If the graph of the function has no holes or breaks in it the function is continuous.

Recently Updated Pages

Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

Give 10 examples for herbs , shrubs , climbers , creepers

Which planet is known as the red planet aMercury bMars class 6 social science CBSE

Which state has the longest coastline in India A Tamil class 10 social science CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE