Solving Limits with Trig Functions

[TommG]

I am having trouble solving limits with trig functions.

lim_x→0 x+xcosx/sinxcosx

useful equation lim_θ→0 sinθ/θ = 1

my attempt

x+xcosx/sinxcosx

x/sinxcosx + xcosx/sinxcosx

x/sinx*1/cosx + x/sinx this is where I am getting stuck

 

[jedishrfu]

the answer is right in front of you.

As x approaches zero then what is the value of cosx?

and you have the sinx/x as x approaches zero so everything is there for you

 

[TommG]

the answer is right in front of you.

As x approaches zero then what is the value of cosx?

and you have the sinx/x as x approaches zero so everything is there for you

Does x/sinx = 1 also?

I thought sinx had to be in the numerator for it to = 1

 

[gopher_p]

Does x/sinx = 1 also?

I thought sinx had to be in the numerator for it to = 1

Technically, you're right. But note that if $x\neq 0$, then $\frac{x}{\sin x}=\frac{1}{\frac{\sin x}{x}}$, and you have something to which you can apply your limit law for ratios.

So I guess technically you're not right (with regard to the value of the limit), but it's good to be skeptical. I had some students a year or so ago that got marked down (per a grading rubric that was not mine) for omitting the step that I showed.