Question

What is the limit as x approaches 0 of (sin(x^2 +1/x)-sin(1/x))/x, step by step, please.

Answer 1

Just to clarify this, is it:
\[\lim_{x \rightarrow \infty} \frac{ \sin(x^2+\frac{ 1 }{ x })-\sin \frac{ 1 }{ x } }{ x }\]

Answer 2

yes it is.

Answer 3

I don't usually suggest this initially, but it will make it a bit easier, so you might want to use L'Hopitals rule. Are you familiar with it?

Answer 4

aproaching to zero

Answer 5

L'Hopitals, no. But I can see on Khan Academy.

Answer 6

Well, first you can start by combining terms on the numerator of \(sin(x^2+..\)

Answer 7

I got it... i will try.

Answer 8

I think you could also use sum-difference formula for sin :)

Answer 9

\[ \lim_{x \rightarrow 0} \frac{ \sin(x^{2} + \frac{ 1 }{ x })-\sin \frac{ 1 }{ x} }{ x } I started with \sin(a+b)= sinx ^{2}\cos \frac{ 1 }{ x }+\sin \frac{ 1 }{ x }cosx ^{2} equals \sin \frac{ 1 }{x} where is my mistake?\]