Question & Answer

Solve the following: $\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin 3x}}{x}$

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Hint: Here we need to convert the given expression into any standard formulae of the limits such that the simplification is easier. Here we will use $\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin x}}{x} = 1$ to evaluate.

Complete step-by-step answer:
We have,
$\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin 3x}}{x}$
Multiply and divide the equation with 3 to simplify the process,
$\mathop {\lim }\limits_{x \to 0} \dfrac{{3\sin 3x}}{{3x}}$
Now, we know that there is a rule which states that,
$\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin x}}{x} = 1$
Therefore, on applying the above formula, we get,
$ \Rightarrow 3 \times 1 = 3$
Answer is 3.

Note: Try to think of a formula which can be applied here so that it is easier to evaluate. Always simplify the given expression in the form of any standard rules/ formulae of limits to solve it easier.