Question

Find the value of $\mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{x}{{\sin x}}} \right] + {x^x}$ where $\left[ . \right]$ is G.I.F.
A.$1$
B.$2$
C.$0$
D.$- 1$

Answer 1

Here, we will find the value of the given expression. We will use the formula for the given trigonometric function and by simplifying the equation and substitute their limits to find the value of the expression. Thus, we will find the value of the given expression.
Formula Used:
$\sin x = x - \dfrac{{{x^3}}}{{3!}} + \dfrac{{{x^5}}}{{5!}} - \dfrac{{{x^7}}}{{7!}}....\infty$

Complete step-by-step answer:
We are given with a function $\mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{x}{{\sin x}}} \right] + {x^x}$
Let us consider $f\left( x \right) = \mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{x}{{\sin x}}} \right] + {x^x}$.
Now, by using the formula $\sin x = x - \dfrac{{{x^3}}}{{3!}} + \dfrac{{{x^5}}}{{5!}} - \dfrac{{{x^7}}}{{7!}}....\infty$, we get
$\Rightarrow f\left( x \right) = \mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{x}{{x - \dfrac{{{x^3}}}{{3!}} + \dfrac{{{x^5}}}{{5!}} - \dfrac{{{x^7}}}{{7!}}....\infty }}} \right] + {x^x}$
Now, by substituting their limits of $x$ to the given functions as $x \to 0$, so we get
$\Rightarrow f\left( x \right) = \mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{0}{0}} \right] + {x^x}$
$\Rightarrow f\left( x \right) = \mathop {\lim }\limits_{x \to 0} 0 + {0^0}$
We know that when any number is raised to the power zero, then the resulting number would be one i.e., ${X^0} = 1$but when zero is raised to the power zero, then the resulting number would be zero.
$\Rightarrow f\left( x \right) = \mathop {\lim }\limits_{x \to 0} 0 + 0$
$\Rightarrow f\left( x \right) = 0$
$\Rightarrow \mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{x}{{\sin x}}} \right] + {X^x} = 0$
Therefore, the value of $\mathop {\lim }\limits_{x \to 0} \left[ {\dfrac{x}{{\sin x}}} \right] + {X^x}$ where $\left[ . \right]$ is G.I.F.is $0$.
Thus, option (C) is the correct answer.

Note: We know that the function that is rounding off the real number down to the integer less than the number is known as the greatest integer function. The greatest integer function is denoted by $f\left( x \right) = \left[ X \right]$ and is defined as the integer less or equal to $x$. This means that if we substitute the variable $x$ as an integer then the answer will be the same integer and if we substitute the variable $x$ as a negative integer then the answer will be the variable which is just before the variable $x$. The collection of values substituted is called the domain of the function and the corresponding values are called the range of the function. Thus, we can check the answer with the limits substituted.