Question

Evaluate Lim sin nx
X>0​

Answer 1

EXPLANATION.

$\sf \implies \lim_{x \to 0} \dfrac{sin(nx)}{x}$

As we know that,

It is a type of trigonometric limits.

one of the most fundamental formula in trigonometric limits is,

$\sf \implies \lim_{x \to 0} \dfrac{sin(x)}{x} = 1$

Multiply and divide numerator and denominator by n, we get.

$\sf \implies \lim_{x \to 0} \dfrac{sin(nx)}{x} \ \times \dfrac{n}{n}$

$\sf \implies \lim_{x \to 0} n.\dfrac{sin(nx)}{nx}$

$\sf \implies n \lim_{x \to 0} \dfrac{sin(nx)}{nx}$

$\sf \implies n \lim_{x \to 0} (1)$

$\sf \implies \lim_{x \to 0} = n$

                                                                                                                   

MORE INFORMATION.

$\sf \implies (A) = \lim_{x \to 0} \dfrac{sin(x)}{x} \sf \implies \lim_{x \to 0} \dfrac{x}{sin(x)} = 1 \ \ Or \sf \implies \lim_{x \to 0} sin(x) = 0$

$\sf \implies (B) = \lim_{x \to 0} cos(x) \sf \implies \lim_{x \to 0} \bigg(\dfrac{1}{cos(x)} \bigg)=1$

$\sf \implies (3) = \lim_{x \to 0} \dfrac{tan(x)}{x} = \sf \implies \lim_{x \to 0} \dfrac{x}{tan(x)} = 1 \ \ Or \sf \implies \lim_{x \to 0} tan(x) = 0$

Answer 2

Given Question :-

$\tt \: Evaluate \: \tt \:\lim_{x\to 0} \: \dfrac{sin \: nx}{x}$

$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{sin \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\large\underline\purple{\bold{Solution :- }}$

$: \longrightarrow \tt \: \tt \:\lim_{x\to 0} \: \dfrac{sin \: nx}{x}$

☆ If we substitute directly x = 0, we get indeterminant form

☆ To evaluate, multiply and divide by n, we get

$: \longrightarrow \tt \: \tt \:\lim_{x\to 0} \: \dfrac{sin \: nx}{n \: x} \times n$

$: \longrightarrow \tt \: \tt \:n \: \lim_{x\to 0} \: \dfrac{sin \: nx}{nx}$

$\tt\implies \:n \: \times \: 1$

$\tt\implies \:n$

$\boxed{ \pink{ \bf \: Hence \: \lim_{x\to 0} \: \dfrac{sin \: nx}{x} = n}}$

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$\large \red{\bf \: ⟼ Explore \: more } ✍$

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{tan \: x}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{ {e}^{x} - 1}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{ log(1 + x)}{x} \: = \: 1 }}}}}} \\ \end{gathered}$

$\begin{gathered}(4)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{ {a}^{x} - 1}{x} \: = \: log(a) }}}}}} \\ \end{gathered}$

$\begin{gathered}(5)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to 0} \: \dfrac{1 - cos \: x}{x} \: = \: 0 }}}}}} \\ \end{gathered}$

$\begin{gathered}(6)\:{\underline{\boxed{\bf{\blue{{\tt \:\lim_{x\to a} \: \dfrac{ {x}^{n} - {a}^{n} }{x - a} \: = \: n {a}^{n - 1} }}}}}} \\ \end{gathered}$