Trigonometric Limits: Factoring and L'Hopital's Rule

[dav1d]

Homework Statement

lim x->0 (2x+1-cosx)/(4x)

Homework Equations

The Attempt at a Solution

factor out 1/4.
get stuck because of cosx..
and I'm not using l'hopital even though I know it.

 

[gb7nash]

Have you learned Taylor series yet?

 

[Mark44]

Homework Statement

lim x->0 (2x+1-cosx)/(4x)

The Attempt at a Solution

factor out 1/4.
get stuck because of cosx..
and I'm not using l'hopital even though I know it.

Have you learned Taylor series yet?

I would bet that the OP hasn't learned Taylor series yet, but you don't need to use them, or L'Hopital's Rule, to evaluate this limit.

Split the limit into two limits, one with (2x)/(4x) and the other with (1 - cosx)/(4x).

There are a couple of special limits that are usually presented in textbooks in sections where there are limit problems involving trig functions. These are
$$\lim_{x \to 0} \frac{sin(x)}{x} = 1$$

and
$$\lim_{x \to 0} \frac{1 - cos(x)}{x} = 0$$

 

[dav1d]

Thanks, I know L'hopital's but I knew there was a easier solution.

By the way, what other special limits are there? Link would be nice!

 

[dynamicsolo]

The two trigonometric limits that Mark44 provided are pretty much the "special" ones that people use. They crop up in developing the derivatives of sin x and cos x using "difference quotients". Those two are useful to know.

The reason people generally don't bother looking for more of these is that once we do have L'Hopital's Rule, we calculate such trigonometric limits using that tool instead...