Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{\sin{4x}}}$

In this limit problem, two trigonometric functions are involved. So, the limit of the trigonometric function can be evaluated by using limit rule of trigonometric function.

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{\sin{4x}}}$

Use Limit Rule for Trigonometric function

There is a limit rule in terms of sin function and it can be used in this limit problem to obtain the limit of this trigonometric function.

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

Let’s try to transform both numerator and denominator in this form.

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg[ \sin{3x} \times \dfrac{1}{\sin{4x}} \Bigg] }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg[ \sin{3x} \times \dfrac{1}{\sin{4x}} \times 1 \Bigg] }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg[ \sin{3x} \times \dfrac{1}{\sin{4x}} \times \dfrac{x}{x} \Bigg] }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg[ \sin{3x} \times \dfrac{1}{\sin{4x}} \times \dfrac{1}{x} \times x \Bigg] }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg[ \sin{3x} \times \dfrac{1}{x} \times \dfrac{1}{\sin{4x}} \times x \Bigg] }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg[ \sin{3x} \times \dfrac{1}{x} \times \dfrac{1 \times x}{\sin{4x}}\Bigg] }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg[ \dfrac{\sin{3x}}{x} \times \dfrac{x}{\sin{4x}}\Bigg] }$

Express the second factor in the function in its reciprocal form.

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Bigg[ \dfrac{\sin{3x}}{x} \times \dfrac{1}{\dfrac{\sin{4x}}{x}}\Bigg] }$

Use Product Rule of Limits

As per the product rule of limits, the limit of the product of trigonometric functions can be written as the product of their limits.

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{x} }$ $\times$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{1}{\dfrac{\sin{4x}}{x}} }$

Use Reciprocal Rule of Limits

According to reciprocal rule of limits, the limit of reciprocal trigonometric function can be written as reciprocal of its limit.

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{x} }$ $\times$ $\dfrac{1}{\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{4x}}{x}} }$

Evaluate the Limit of each trigonometric function

The limit rule of trigonometric function cannot be applied at this time because the angle in the sine function should also be in its denominator. So, let’s try to repeat the same technique.

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{x} \times 1 }$ $\times$ $\dfrac{1}{\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{4x}}{x} \times 1} }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{x} \times \dfrac{3}{3} }$ $\times$ $\dfrac{1}{\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{4x}}{x} \times \dfrac{4}{4}} }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x} \times 3}{x \times 3}}$ $\times$ $\dfrac{1}{\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{4x} \times 4}{x \times 4}} }$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{3\sin{3x}}{3x}}$ $\times$ $\dfrac{1}{\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{4\sin{4x}}{4x}} }$

Now, use constant multiple rule of limits to separate the constants from functions.

$= \,\,\,$ $3 \times \displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{3x}}$ $\times$ $\dfrac{1}{4 \times \displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{4x}}{4x}} }$

If $x \to 0$, then $3x \to 3 \times 0$ and $4x \to 4 \times 0$. Therefore, $3x \to 0$ and $4x \to 0$. Therefore, if $x$ approaches $0$, then $3x$ and $4x$ also approach to $0$.

$= \,\,\,$ $3 \times \displaystyle \large \lim_{3x \,\to\, 0}{\normalsize \dfrac{\sin{3x}}{3x}}$ $\times$ $\dfrac{1}{4 \times \displaystyle \large \lim_{4x \,\to\, 0}{\normalsize \dfrac{\sin{4x}}{4x}} }$

According to limit of sin(x)/x as x approaches 0 formula, the limit of each function is equal to $1$.

$= \,\,\,$ $3 \times 1$ $\times$ $\dfrac{1}{4 \times 1}$

$= \,\,\,$ $3$ $\times$ $\dfrac{1}{4}$

$= \,\,\,$ $\dfrac{3 \times 1}{4}$

$= \,\,\,$ $\dfrac{3}{4}$