Question

The limit of (sin^3)x/x as x approaches 0

Answer 1

This:
\[\Large \lim_{x\rightarrow 0}\frac{\sin^3(x)}{x}\]

Answer 2

well yeah

Answer 3

Did you know that the limit of a product is the product of the limits... if both limits exist? :D
Why don't we make a little adjustment:
\[\Large \lim_{x\rightarrow 0}\frac{\sin(x)}{x}\cdot \sin^2(x)\]
Does any bit of this look familiar to you?

Answer 4

\[\frac{ sinx }{x }\]\[=1\]

Answer 5

That's right :)
What about the limit of \[\large \sin^2(x)\] as x goes to zero? The sine function is continuous after all...

Answer 6

So what about \[\sin ^{2}(x)\] ?

Answer 7

It's continuous, so the limit of that as x goes to zero is just \(\large \sin^2(x)\) itself evaluated at x = 0

Answer 8

alright that makes more sense. Thank you!

Answer 9

So what's your answer?

Answer 10

Isn't it 0?

Answer 11

Wait...

Answer 12

Yeah it's not coming to me

Answer 13

Okay
\[\Large \lim_{x\rightarrow 0}\cancel{\color{blue}{\frac{\sin(x)}{x}}}^1\cdot \sin^2(x)\]This bit goes to 1 and...
\[\Large \lim_{x\rightarrow 0}\cancel{\color{blue}{\frac{\sin(x)}{x}}}^1\cdot \cancel{\color{red}{\sin^2(x)}}^0\]
This bit goes to zero... so all in all?

Answer 14

it leaves \[\sin ^{2}x\]

Answer 15

Yeah...which goes to...?

Answer 16

0