lim (tanx - sinx / x^3 ) = ?
x->0

I tried to do lopital , but, I did was stuck somewhere... I got 1/4.. the

Is that \lim_{x\to 0}\frac{tan(x)-sin(x)}{x^{3}} or what you have typed:

\lim_{x\to 0}[tan(x)-\frac{sin(x)}{x^{3}}]

I assume the former. Grouping symbols are important.

You can use L'Hopital, but it requires several applications.

First application:

\frac{-cos(x)+2tan(x)sec^{2}(x)}{3x^{2}}

second application:

\frac{sin(x)+2tan(x)sec^{2}(x)}{6x}

Third application:

\frac{cos(x)+2sec^{4}(x)+4tan^{2}(x)sec^{2}(x)}{6}

Now, letting x=0 in the numerator, we get:

\frac{3}{6}=\frac{1}{2}


Here is a non-L'Hopital way to go about it. I like this better.

Rewrite as:

\lim_{x\to 0}\frac{tan(x)(1-cos(x))}{x^{3}}

=\lim_{x\to 0}\frac{tan(x)}{x}\cdot \frac{1-cos(x)}{x^{2}}

Now, take each limit separately.

\lim_{x\to 0}\frac{tan(x)}{x}=\lim_{x\to 0}\left(\frac{sin(x)}{x}\right)\left(\frac{1}{cos( x)}\right)=1\cdot 1=1

Now for:

\lim_{x\to 0}\frac{1-cos(x)}{x^{2}}

=\lim_{x\to 0}\frac{1-cos(x)}{x^{2}}\cdot \frac{1+cos(x)}{1+cos(x)}

=\lim_{x\to 0}\frac{1-cos^{2}(x)}{x^{2}(1+cos(x))}

=\lim_{x\to 0}\frac{sin^{2}(x)}{x^{2}}\cdot \frac{1}{1+cos(x)}

=(1)(\frac{1}{2})=\frac{1}{2}

From the first limit we have 1. So, we get (1)(1)(\frac{1}{2})

Thus, the limit is 1/2

Note that \lim_{x\to 0}\frac{sin(x)}{x}=1 is a famous limit used often when proving other limits. It is not necessary to prove it here, but use it as a lemma, so to speak. It's proof usually involves the Squeeze Theorem.

As for \lim_{x\to 0}\frac{1}{1+cos(x)}, we can use an approximation. Knowing that cos(x)\approx 1-\frac{x^{2}}{2} for x near 0, we can sub this in and get:

\lim_{x\to 0}\frac{2}{4-x^{2}}.

Now, as can be seen, as x\to 0, we are left with 1/2.

Actually, we could have used this approximation, as well as sin(x)\approx x for x near 0, to prove from the outset.

Doing so, leads to

\lim_{x\to 0}\frac{tan(x)-sin(x)}{x^{3}}=\lim_{x\to 0}\frac{\frac{sin(x)}{cos(x)}-sin(x)}{x^{3}}

=\lim_{x\to 0}\frac{\frac{x}{1-\frac{x^{2}}{2}}-x}{x^{3}}=\lim_{x\to 0}\frac{1}{2-x^{2}}

Now, as x\to 0, we are left with... again... 1/2.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

When dealing with trig limits with x\to 0, then the following approximations will prove useful.

1-\frac{x^{2}}{2}\approx cos(x)

x\approx sin(x)

\frac{2-x^{2}}{2x}\approx tan(x)

\frac{2x}{2-x^{2}}\approx cot(x)

\frac{2}{2-x^{2}}\approx sec(x)

\frac{1}{x}\approx csc(x)

These are known as 'asymptotic' approximations.