How can I solve this limit without using d'hospital method?

[player1_1_1]

Homework Statement

hello, what can I do to solve this without using d'hospital method?
$$\lim_{x\to0}\frac{\tan x-x}{x^3}$$

 

[Sweet_GirL]

Homework Statement

hello, what can I do to solve this without using d'hospital method?
$$\lim_{x\to0}\frac{\tan x-x}{x^3}$$

Recall the formula [which is just the M.S. for tan(x)]:
$$tan(x)=x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+...$$
Substitute this in the limit and the two "x" will be cancelled.
Then divide the denominator and the numerator by x^3.
and the limit will be done by the direct substitution.
The limit = 1/3

 

[cap.r]

look up the taylor series for sin and cosine and write the one for tan. Now factor out X^3 from the top and take that limit as X goes to Zero