Answer

Verified

365.7k+ views

**Hint:**To solve this question we need to have the knowledge of limits. To solve the problem where the function given is $x\csc x$, we will first convert the trigonometric function$\csc x$ in terms of $\sin x$ and then we will be applying the value of the limit which is given to us as $x$ tends to zero.

**Complete step by step answer:**

The question asks us to find the value of the function$\displaystyle \lim_{x \to 0}\left( x\csc x \right)$. We are given with a function which is a product of the algebraic function $x$ and the trigonometric function$\csc x$. The first step is to change the trigonometric function $\csc x$ into the terms of $\sin x$. So as you know, the trigonometric function $\csc x$ is the reciprocal of the trigonometric function $\sin x$. Mathematically it will be written as:

$\Rightarrow \csc x=\dfrac{1}{\sin x}$

$\Rightarrow x\csc x=x\dfrac{1}{\sin x}$

$\Rightarrow x\csc x=\dfrac{x}{\sin x}$

Now to find the value of $\displaystyle \lim_{x \to 0}\left( x\csc x \right)$which has become as $\displaystyle \lim_{x \to 0}\left( \dfrac{x}{\sin x} \right)$ we will divide both the numerator and the denominator of the function by $x$ , on doing this we get:

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

On checking the numerator of the function we see that it cancels to give the result as $1$ as $\dfrac{x}{x}$ is $1$. Now on applying the limit to the function $\dfrac{\sin x}{x}$. As we know that $\displaystyle \lim_{x \to 0}\left( \dfrac{\sin x}{x} \right)=0$, so applying this in the above expression we get:

$\Rightarrow \displaystyle \lim_{x \to 0}\left( \dfrac{1}{\dfrac{\sin x}{x}} \right)$

$\Rightarrow \dfrac{1}{1}$

Since any fraction having denominator as $1$, then the numerator of the fraction changes to an integer. So on doing this we get:

$\Rightarrow 1$

$\therefore $ The value of $\displaystyle \lim_{x \to 0}\left( x\csc x \right)$ is $1$.

**Note:**To solve the problem like this we need to first solve the function and then need to substitute the value of the limit to the function. When the fraction has a denominator as $1$, then the numerator of the fraction changes to an integer.

Recently Updated Pages

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Advantages and disadvantages of science

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference Between Plant Cell and Animal Cell

Which are the Top 10 Largest Countries of the World?

10 examples of evaporation in daily life with explanations

Give 10 examples for herbs , shrubs , climbers , creepers

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

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