Answer

Verified

414.9k+ views

**Hint:**First check if the given limit is in the form ${{\left( 0 \right)}^{0}}$ or not. For ${{\left( 0 \right)}^{0}}$ form simplify using the formula $\displaystyle \lim_{x \to 0}{{\left( f\left( x \right) \right)}^{g\left( x \right)}}={{e}^{\displaystyle \lim_{x \to 0}g\left( x \right)\ln \left( f\left( x \right) \right)}}$. After simplification, put the value of ‘x’ as ‘0’ and do the necessary calculations to get the limiting value.

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

Putting the value of ‘x’ in the function we are getting ${{\left( \sin 0 \right)}^{0}}={{\left( 0 \right)}^{0}}$

The expression of the form $\displaystyle \lim_{x \to 0}{{\left( f\left( x \right) \right)}^{g\left( x \right)}}$ with the value ${{\left( 0 \right)}^{0}}$ can be simplified by taking as ${{e}^{\displaystyle \lim_{x \to 0}g\left( x \right)\ln \left( f\left( x \right) \right)}}$

Considering our equation $\displaystyle \lim_{x \to {{0}^{+}}}{{\left( \sin x \right)}^{x}}$

By comparison, $f\left( x \right)=\sin x$ and $g\left( x \right)=x$

So, it can be simplified as

$\Rightarrow {{e}^{\displaystyle \lim_{x \to {{0}^{+}}}x\ln \left( \sin x \right)}}$

Taking the value of ‘x’ as ‘0’, we get

$x\ln \left( \sin x \right)=0\times \ln \left( \sin x \right)=0$

Putting this value in the equation, we get

\[\begin{align}

& \Rightarrow {{e}^{0}} \\

& \Rightarrow 1 \\

\end{align}\]

Hence, $\displaystyle \lim_{x \to {{0}^{+}}}{{\left( \sin x \right)}^{x}}=1$

Where, ‘1’ is the limiting value of the given limit.

**This is the required solution of the given question.**

**Note:**We know in logarithmic function the value of log which we are taking must be greater than ‘0’. So, in $\ln \left( \sin x \right)$, the value of $\sin x$ must be greater than ‘0’. Again as we know the range of $\sin x$ is $\left[ -1,1 \right]$, but since we have to get only those values which are greater than ‘0’, so now the range becomes $\left( 0,1 \right]$. Hence, multiplying this range of $\ln \left( \sin x \right)$ with ‘0’ we are getting $x\ln \left( \sin x \right)=0\times \ln \left( \sin x \right)=0$ as the value of $\sin x$ with range $\left( 0,1 \right]$ will always be a positive value.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Who was the Governor general of India at the time of class 11 social science CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

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

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference Between Plant Cell and Animal Cell