Evaluate $\displaystyle \lim_{x\,\to\,\infty}{\normalsize x\sin{\Big(\dfrac{\pi}{x}\Big)}}$ by Limit rules

The limit of the product of $x$ and sine of angle $\pi$ divided by $x$ should be evaluated by using the fundamental limit rules in this problem as the value of $x$ approaches infinity. It is evaluated that the limit of $x$ times sine of angle $\pi$ divided by $x$ as the value of $x$ tends to infinity is indeterminate.

$\displaystyle \lim_{x\,\to\,\infty}{\normalsize \bigg(x \times \sin{\Big(\dfrac{\pi}{x}\Big)}\bigg)}$ $\,=\,$ $\infty \times 0$

It clears that the direct substitution method is not useful to find the limit of $x$ times sine of $\pi$ divided by $x$ as the value of $x$ approaches infinity. So, let us try to learn how to find the limit by using the math formulas.

Express the function as a Rational function

The given function is a product of an algebraic function and a trigonometric function. The algebraic function $x$ is also there in reciprocal form as a part of angle inside the sine function. There is a trigonometric limit rule in sine function. So, it is a good idea to convert the algebraic function $x$ as the reciprocal of a reciprocal rule.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \bigg(\dfrac{1}{\Big(\dfrac{1}{x}\Big)} \times \sin{\Big(\dfrac{\pi}{x}\Big)}\bigg)}$

Now, multiply the functions to obtain their product as per the product rule of the fractions, and it helps us to convert the product of the two functions as a rational function.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{1 \times \sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{1}{x}\Big)}}$

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{1}{x}\Big)}}$

Convert the function to use Trigonometric Limit rule

Observe the rational function closely. The angle inside the sine function is $\pi$ divided by $x$ and the reciprocal of $x$ is there in the denominator. So, the rational function in limit operation approximately matches with the trigonometric limit rule. Now, let us try to convert the function same as the trigonometric limit rule in sine function and it can be done by including the pi in the expression of denominator.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{1 \times \dfrac{1}{x}}}$

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\dfrac{\pi}{\pi} \times \dfrac{1}{x}}}$

It is time to make the expression in the denominator same as the function inside the sine in the numerator. It can be done by using the multiplication rule of the fractions.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\dfrac{1 \times \pi}{\pi} \times \dfrac{1}{x}}}$

Now, split the first factor in the denominator as a product of two factors and the expression is written as a product of three factors.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\dfrac{1}{\pi} \times \pi \times \dfrac{1}{x}}}$

Now, multiply the second and third factors by the multiplication rule of the fractions and get their product.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\dfrac{1}{\pi} \times \dfrac{\pi \times 1}{x}}}$

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\dfrac{1}{\pi} \times \dfrac{\pi}{x}}}$

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\dfrac{1}{\pi} \times \Big(\dfrac{\pi}{x}\Big)}}$

The second factor in the expression of the denominator is exactly same as the function inside the sine function in numerator but the reciprocal of $\pi$ is there additionally in the denominator. So, it should be separated from the rational function to use the trigonometric limit rule.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{1 \times \sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{1}{\pi}\Big) \times \Big(\dfrac{\pi}{x}\Big)}}$

According to the multiplication of the fractions, the rational function can be now split as a product of two rational functions.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \bigg(\dfrac{1}{\Big(\dfrac{1}{\pi}\Big)} \times \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{\pi}{x}\Big)}\bigg)}$

The multiplicative inverse of the reciprocal of $\pi$ is equal to $\pi$ as per the reciprocal rule of the reciprocal function.

$=\,\,$ $\displaystyle \lim_{x\,\to\,\infty}{\normalsize \bigg(\pi \times \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{\pi}{x}\Big)}\bigg)}$

The symbol pi represents an irrational constant value. So, it can be taken out from the limit operation as per the constant multiple rule of limits.

$=\,\,$ $\pi \times \displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{\pi}{x}\Big)}}$

Find the Limit by Trigonometric Limit formula

The function inside the sine function is exactly same as the function in the denominator, and the limit of the rational function can be evaluated by the trigonometric limit rule in sine function but the input of the limit operation should also be converted.

$(1).\,\,$ If $x\,\to\,\infty$, then $\dfrac{1}{x}\,\to\,\dfrac{1}{\infty}$. Therefore, $\dfrac{1}{x}\,\to\,0$

$(2).\,\,$ If $\dfrac{1}{x}\,\to\,0$, then $\pi \times \dfrac{1}{x}\,\to\,\pi \times 0$. Therefore, $\dfrac{\pi}{x}\,\to\,0$

The two steps have cleared that pi divided by $x$ approaches $0$ as the value of $x$ tends to infinity.

$\implies$ $\pi \times \displaystyle \lim_{x\,\to\,\infty}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{\pi}{x}\Big)}}$ $\,=\,$ $\pi \times \displaystyle \lim_{{\Large \frac{\pi}{x}}\,\to\,0}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{\pi}{x}\Big)}}$

Now, let us find the limit of the sine of angle $\pi$ divided by $x$, divided by $\pi$ divided $x$ as the value of $\pi$ divided by $x$ approaches $0$. Now, let’s denote the $\pi$ divided by $x$ by a variable $y$.

$\implies$ $\pi \times \displaystyle \lim_{{\Large \frac{\pi}{x}}\,\to\,0}{\normalsize \dfrac{\sin{\Big(\dfrac{\pi}{x}\Big)}}{\Big(\dfrac{\pi}{x}\Big)}}$ $\,=\,$ $\pi \times \displaystyle \lim_{y\,\to\,0}{\normalsize \dfrac{\sin{(y)}}{(y)}}$

It is time to evaluate the expression on the right hand side of the equation.

$=\,\,$ $\pi \times \displaystyle \lim_{y\,\to\,0}{\normalsize \dfrac{\sin{y}}{y}}$

According to the trigonometric limit rule, the sine of angle $y$ divided by $y$ as the value of $y$ approaches $0$ is equal to one.

$=\,\,$ $\pi \times 1$

$=\,\,$ $\pi$