Question

Derivatives!
How did we end up getting this answer? Could you please show me some helpful steps?

If limit as x approaches 0, what is the derivative of (7 * sin 7 x)/(7 * 4x) = 7/4

Answer 1

My own attempt:
7 * sin 7(0) / 7 * 4(0) = 7 * 0 / 7 * (0) = 0/0 = infinity because limit does not mean exactly at a point, it is almost that point.

Answer 2

So you're asked to evaluate a limit?
This doesn't look like it's connected to looking for a derivative :)
Maybe you're confused on that hehe.

It makes use of this limit identity:\[\large\rm \lim_{x\to0}\frac{\sin x}{x}=1\]Have you seen this before?

Answer 3

Yes, we are asked to evaluate the limit of this. However, we are also studying Derivatives at school, so I thought it was derivatives :D .Right yes, I have seen that equation.

So the answer is:

7 * 7 sin x / 7 * 4 * x = 7 * 7 / 7 * 4 = 7/ 4 Am I right?

Answer 4

Oh boy it's hard to read that in text +_+
But no, you can't pull a 7 out of sine like that.

Notice that the important thing about our identity is that the `angle` (the thing inside of the sine) must match the `denominator`. It doesn't have to be exactly x though, it could be anything.

So we can use our identity to say,\[\large\rm \lim_{x\to0}\frac{\sin4x}{4x}=1\]Ok maybe we can make use of this...

Answer 5

Ahh woops, I meant to say,\[\large\rm \lim_{x\to0}\frac{\sin7x}{7x}=1\]Maybe we can make use of that ^

Answer 6

Because the 7x is trapped inside of the sine operation,
we won't be able to change that.

So maybe we can change our denominator into a 7x somehow.\[\large\rm \lim_{x\to0}\frac{7\sin7x}{7\cdot4x}\]

Answer 7

Ok that makes sense, so the angles must be equal which is 7X on numerator and denominator.

Answer 8

Remember that multiplication is commutative,
we can rearrange the order in which we multiply things.
So maybe we switch the 7 and 4 in the denominator multiplication,\[\large\rm \lim_{x\to0}\frac{7\sin7x}{4\cdot7x}\]That's looking a little bit closer to what we want, yes?

Answer 9

Exactly, now sin 7x / 7x becomes 1 . Which ultimately results in 7/ 4

Answer 10

It makes the problem look a little bit nicer if you take one extra step and pull the "junk" out to the front of the limit before applying the identity,\[\large\rm \lim_{x\to0}\frac{7\sin7x}{4\cdot7x}\quad=\quad \frac{7}{4}\lim_{x\to0}\frac{\sin7x}{7x}\]But yes :)

Answer 11

Haha yes, that is the constant rule in limits. We can do that for complicated equations.

Answer 12

yayyy team, we did it! :D

Answer 13

Thank you zepdrix, awsome teamwork. :D