What is the derivative of a function represented by lim sin x / x at x=0?

[Mathnewbie]

Good Evening

I am studying for my exam next week and am not sure how to answer this question from one of my term test.

lim sin x / x represents the derivative of what function at what
x->0

number?

Does anybody know to do this?

thanks

 

[hypermorphism]

Review the definition of the derivative at a domain point.

 

[benorin]

$$\lim_{x\rightarrow 0}\frac{\sin x}{x}=\lim_{x\rightarrow 0}\frac{\cos x}{1}=1$$

where the first equality is by l'Hospital's rule.

 

[matt grime]

The question doesn't ask you to evaluate it (and using l'hopital is implictly forbidden) it merely asks what this limit is in the sense of identifying it.

from the definition of derivative we get the answer to the actual question asked.

 

[Mathnewbie]

well the definition of the derivative is

f(a+h) -f(a)
f'(a) = lim -------------
h->0 h

not sure how this helps me. Just having problems with this stuff. look through my notes and textbook and cannot find any problems like this.

Not sure how to handle this.

 

[shmoe]

You need to find a specific function f, and a corresponding point $$a$$ so that the limit definition you gave of f'(a) is the same as the limit you started with.

 

[Mathnewbie]

oh Man. is it Sin x at 0?

 

[matt grime]

and the light bulb goes on.

 

[ovoleg]

I remember seeing a proof for this but that was one of the identities we used before we learned L'Hospitals rule...

Lim x->0 (Sinx/x) = 1

Useful for problems such as lim x->0 Sin7x/3x and so on, without knowing L'Hospitals rule.

My calc 1 instructor would just give zero if you used L'Hospitals rule on any of these problems