Question

lim x tends to 0 sin5x/tan3x

Answer 1

\[\large \lim_{x \rightarrow 0}\frac{\sin5x}{\tan3x} \qquad = \qquad \lim_{x \rightarrow 0}\frac{\sin5x}{\sin3x}\cos3x\]Understand what I did so far?
I just rewrote tangent as sines and consines using the appropriate identity.

Answer 2

Here is an important identity that we can use,\[\large \lim_{\theta \rightarrow 0}\frac{\sin \theta}{\theta} \qquad = \qquad 1\]

Answer 3

the answer is 5/3 and i dnt know how do i get that

Answer 4

Yes I'm trying to explain that... Are you following along..?

Answer 5

yeah i am

Answer 6

Is it powers, or coefficients?

Answer 7

Umm I can't think of the right way to explain this. It's a bit of a tricky problem.

We're going to split into a product of two limits. and we need to form a 5x on the bottom, below the sin5x. And a 3x above the sin3x so we can apply our identity.

This might be a little confusing, see if you can follow these steps.

Answer 8

coefficients

Answer 9

nope i dint

Answer 10

Multiply the top and bottom by x, 5, and 3,
\[\large \lim_{x \rightarrow 0}\frac{5\cdot3x\sin5x}{3\cdot5x\sin3x}\cos3x\]

Which we can write like this,
\[\large \dfrac{5}{3}\lim_{x \rightarrow 0}\frac{3x\sin5x}{5x\sin3x}\cos3x\]

Answer 11

Whoops...

Answer 12

\[\large \dfrac{5}{3}\lim_{x \rightarrow 0}\frac{\sin5x}{5x}\cos3x\cdot \lim_{x \rightarrow 0}\frac{3x}{\sin3x}\]

Answer 13

I did a bunch of shuffling things around. Like I said, this one is a bit tricky :\ you have to be comfortable with your multiplication.

Answer 14

what happens to cos 3x ?

Answer 15

Based on the identity posted earlier, we know that these orange parts will give us 1.\[\large \dfrac{5}{3}\color{orangered}{\lim_{x \rightarrow 0}\frac{\sin5x}{5x}}\cos3x\cdot \color{orangered}{\lim_{x \rightarrow 0}\frac{3x}{\sin3x}}\]
And luckily the cosine is also approaching 1 as x approaches 0.
So we end up with, \(\large \dfrac{5}{3}\cdot\color{orangered}{1}\)

Answer 16

You can't pull the cosine outside of the limit, since it includes x.
But as long as we take the limit of it, we're fine.

Cosine 0 is 1 yes? :D

Answer 17

yea thanks a lot