Evaluate $\displaystyle \large \lim_{x\,\to\,\infty}{\normalsize x\sin{\bigg(\dfrac{1}{x}\bigg)}}$ by the Change of variable

The limit of the product of $x$ and sine of angle the reciprocal of $x$ should be evaluated as the value of $x$ approaches infinity by using the limit rules in this limit problem.

Find the Limit by the Direct substitution

Firstly, let us try to find the limit of $x$ times sine of angle one divided by $x$ by using the direct substitution, and it is evaluated that the limit is indeterminate.

$\implies$ $\displaystyle \large \lim_{x\,\to\,\infty}{\normalsize x\sin{\bigg(\dfrac{1}{x}\bigg)}}$ $\,=\,$ $\infty \times 0$

The indeterminate form expresses that the direct substitution method is not useful to find the limit of the product of $x$ and sine of angle $1$ divided by $x$. So, it should be evaluated in another method.

Eliminate indeterminate form by change of variable

The variable $x$ is a main reason for the indeterminate form but its reciprocal does not play any role in it. So, representing the multiplicative inverse of $x$ by a variable is a better idea to remove the indeterminate form. Let’s denote the reciprocal of $x$ by a variable $y$.

If $x\,\to\,\infty$, then $\dfrac{1}{x}\,\to\,\dfrac{1}{\infty}$. Therefore $\dfrac{1}{x}\,\to\,0$. We have considered that the reciprocal of $x$ is denoted by a variable $y$. Therefore, $y\,\to\,0$.

Now, the mathematical expression in terms of $x$ can be written as a mathematical expression in terms of $y$ by the changing the variable technique.

$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \large \lim_{x\,\to\,\infty}{\normalsize x\sin{\bigg(\dfrac{1}{x}\bigg)}}$ $\,=\,$ $\displaystyle \large \lim_{y\,\to\,0}{\normalsize \dfrac{1}{y} \times \sin{y}}$

Find the Limit of function by a Limit formula

Now, multiply the reciprocal of $y$ by the sine of angle $y$ as per the multiplication of fractions.

$=\,\,$ $\displaystyle \large \lim_{y\,\to\,0}{\normalsize \dfrac{1 \times \sin{y}}{y}}$

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

It is time to find the limit of the sine of $y$ divided by $y$ as the value of $y$ approaches zero. According to the trigonometric limit rule in sine function, the limit of sine of $y$ divided by $y$ as $y$ tends to $0$ is equal to one.

$=\,\, 1$

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