Try this Limiting Sum!

Calculus Level 1

$\large \lim_{x\to0} \dfrac x{\sin x} =\, ?$

\infty

1 0 Not Defined

3 solutions

Sam Bealing
May 20, 2016

The limit goes to $\dfrac{0}{0}$ so we may use L'Hopital's Rule:

$\lim_{x \rightarrow 0} \dfrac{\sin{x}}{x}=\lim_{x \rightarrow 0} \dfrac{\cos{x}}{1}=\boxed{1}$

Moderator note:

Simple standard approach.

Samara Simha Reddy
May 20, 2016

$\Large \displaystyle \color{#D61F06}{\lim_{x \rightarrow 0}} \frac{\color{#3D99F6}{\sin x}}{\color{#20A900}{x}} = \color{#D61F06}{\lim_{x \rightarrow 0}} \frac{\color{#3D99F6}{x}}{\color{#20A900}{\sin x}} = \color{#EC7300}{1}$

Moderator note:

Why is the first equality true?

You can't just flip a limit upside down because it goes to $\dfrac{0}{0}$ :

$\lim_{x \rightarrow 0} \dfrac{x^3}{x} \neq \lim_{x \rightarrow 0} \dfrac{x}{x^3}$

It's better to apply L'Hopital's Rule.

Sam Bealing - 5 years ago

come to slack.

Ayush G Rai - 5 years ago

Nope I m leaving!

Samara Simha Reddy - 5 years ago

Lol! Look this question.

Ashish Menon - 5 years ago

Yeah! I got inspired by your problem $\Large \ddot\smile$

Samara Simha Reddy - 5 years ago

Hahaha thanks!

Ashish Menon - 5 years ago

@Ashish Menon – $\Large \ddot\smile$

Samara Simha Reddy - 5 years ago

Vaibhav Vinayaka
Jun 21, 2018

Lim as x-->0 x/sinx = lim as x-->0 1/(sinx/x)= 1/1 = 1

Try this Limiting Sum!

3 solutions

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