Understanding a Step in Finding a Limit: Lim(x->0) x.sqrt(x+2) / sin(x)

[Cal124]

Apologies if this is in the wrong place. I'm struggling to understand a step in finding a limit
Lim(x->0) x.sqrt(x+2) / sin(x)
Following the given solution I get to the point where it's all divided through by x to give
Sqrt(x+2) / sin x/x
Which as approaching 0 gives
Sqrt(2) / 1 = sqrt(2)
I'm struggling how sin x/x is 1
Any help would be great and any advice on limits or links to resources would be appreciated!

 

[axmls]

The limit as sin x/x approaches 0 being 1 is usually a standard result that's done in calculus 1 courses--i.e. In most courses I've seen, the professor derives that in class, usually using the squeeze theorem, and then the students usually memorize it.

Otherwise, if you know l'Hôpital's rule, then you can use that. I'm not aware of any other methods.

 

[Cal124]

The limit as sin x/x approaches 0 being 1 is usually a standard result that's done in calculus 1 courses--i.e. In most courses I've seen, the professor derives that in class, usually using the squeeze theorem, and then the students usually memorize it.

Otherwise, if you know l'Hôpital's rule, then you can use that. I'm not aware of any other methods.

Thanks! Just clicked into place now, remember him explaining this now. There is some frustrating gaps in my knowledge due to a non traditional route to higher education.
Thanks again!

 

[mathwonk]

if you know the derivative of sin is cos, then the limit of sin(x)/x as x-->0 is the value of the derivative of sin at x=0, i.e. cos(0) = 1. of course most people prove this using the value of that limit, although there are other approaches, such as defining sin and cos as solutions of a certain differential equation, or by giving their taylor series. my detailed explanation of the limit approach is here:

http://alpha.math.uga.edu/~roy/tangents_to_y.pdf

 

[Cal124]

if you know the derivative of sin is cos, then the limit of sin(x)/x as x-->0 is the value of the derivative of sin at x=0, i.e. cos(0) = 1. of course most people prove this using the value of that limit, although there are other approaches, such as defining sin and cos as solutions of a certain differential equation, or by giving their taylor series. my detailed explanation of the limit approach is here:

http://alpha.math.uga.edu/~roy/tangents_to_y.pdf

That's a much easier way to grasp the concept, thanks! Will get reading!
Sorry, I don't suppose you have any advice/similar PDF on the definition of a limit? I feel I'm getting there (although part of me feels it could all be wrong and I'm miles away)

 

[MidgetDwarf]

That's a much easier way to grasp the concept, thanks! Will get reading!
Sorry, I don't suppose you have any advice/similar PDF on the definition of a limit? I feel I'm getting there (although part of me feels it could all be wrong and I'm miles away)

You mean the episilon/delta?

 

[Cal124]

You mean the episilon/delta?

Yeah