Find $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sin{5x}-\sin{3x}}{x}}$

The limit of the quotient of the subtraction of sine of angle three times $x$ from the sine of angle five times $x$ divided by $x$ as $x$ approaches to zero is written in the following mathematical form.

$\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sin{5x}-\sin{3x}}{x}}$

Firstly, let us try to calculate the limit of sine of angle $5x$ minus sin of angle $3x$ divided by $x$ as $x$ tends to $0$ by the direct substitution method.

$=\,\,\,$ $\dfrac{\sin{\big(5(0)\big)}-\sin{\big(3(0)\big)}}{0}$

$=\,\,\,$ $\dfrac{\sin{(5 \times 0)}-\sin{(3 \times 0)}}{0}$

$=\,\,\,$ $\dfrac{\sin{(0)}-\sin{(0)}}{0}$

According to the trigonometry, the sine of angle zero radian is zero. Now, substitute it in the above trigonometric rational expression.

$=\,\,\,$ $\dfrac{0-0}{0}$

$=\,\,\,$ $\dfrac{0}{0}$

It is calculated the limit of sine of angle $5$ times $x$ minus sine of angle $3$ times $x$ divided by $x$ as tends to zero is indeterminate as per the direct substitution method. So, it is not recommendable mathematical approach to find the limit but it can be evaluated in three different methods.

Now, let us learn how to find the limit of the ratio of sine of angle $5x$ minus sine of angle $3x$ by $x$ as the value of $x$ is closer to $0$ in each method easily.