Limit of Trig Ratio: limit of x/sin x as x approaches 0

[OldMan]

I am trying to find the limit as x approaches 0 of x/sin x.
I've been trying to either get to the reciprocal which I know is 1, or get something better in the denominator using identities.
But I haven't been able to figure out a way to use identities to get anything other than 0 in the denominator.
Any insights or a little push in the right direction will be greatly appreciated.
Thank you

 

[Deleted member 4993]

Re: Limit of Trig Ratio

I am trying to find the limit as x approaches 0 of x/sin x.
I've been trying to either get to the reciprocal which I know is 1, or get something better in the denominator using identities.
But I haven't been able to figure out a way to use identities to get anything other than 0 in the denominator.
Any insights or a little push in the right direction will be greatly appreciated.
Thank you

\(\displaystyle \lim_{x \rightarrow a}{\frac{1}{f(x)}} \, = \, \frac{1}{\lim_{x \rightarrow a}f(x)}\)

 

[OldMan]

Re: Limit of Trig Ratio

Thank you for the hint but unfortunately I have not been able to get what I am told is the correct answer which is 1.
I still can't get 0 out of the denominator.
Appreciate the help and patience

 

[BigGlenntheHeavy]

Re: Limit of Trig Ratio

OldMan, are you familiar with the Marquis de L'Hopital?

 

[OldMan]

Re: Limit of Trig Ratio

I am now. Thank you.

 

[Deleted member 4993]

Re: Limit of Trig Ratio

I am trying to find the limit as x approaches 0 of x/sin x.
I've been trying to either get to the reciprocal which I know is 1, or get something better in the denominator using identities.
But I haven't been able to figure out a way to use identities to get anything other than 0 in the denominator.
Any insights or a little push in the right direction will be greatly appreciated.
Thank you

\(\displaystyle \lim_{x to a}{\frac{1}{f(x)}} \, = \, \frac{1}{\lim_{x to a}{f(x)}}\)

\(\displaystyle \lim_{x to 0}{\frac{x}{\sin(x)} \, = \, \lim_{x to 0}{\frac{1}{\frac{\sin(x)}{x}} \, = \, \frac{1}{\lim_{x to 0}{\frac{\sin(x)}{x}}} \, = \, \frac{1}{1} \, = \, 1\)