1. ## Limit

Evaluate

$

\lim_{x\to0} \frac {(e^x)^2-1-x^2-\frac {x^4}{2}}{cosx-1+\frac {x^2}{2}-\frac {x^4}{4!}}
$

Thanks.

2. Originally Posted by khatz
Evaluate

$

\lim_{x\to0} \frac {e^x-1-x^2-\frac {x^4}{2}}{cosx-1+\frac {x^2}{2}-\frac {x^4}{4!}}
$

Thanks.
When you take this limit the way it is, you end up with the indeterminate case $\frac{0}{0}$

So you can now apply L'Hôpital's rule.

Thus, $\lim_{x\to0} \frac {e^x-1-x^2-\frac {x^4}{2}}{cosx-1+\frac {x^2}{2}-\frac {x^4}{4!}}=\lim_{x\to0} \frac {e^x-2x-2x^3}{-\sin x-x-\frac {x^3}{3!}}$

However, when we evaluate the limit now, it approaches $\frac{1}{0}$, which is undefined.

Does this help?

3. Take taylor expand:
$e^{x}=1+\frac{x}{1!}+\frac{x^{2}}{2!}+...+\frac{x^ {n}}{n!}+o(x^{n})
cos(x)=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-...+(-1)^{n}\frac{x^{2n}}{(2n)!}+o(x^{2n+1})$

$\lim_{x\to0} \frac {e^x-1-x^2-\frac {x^4}{2}}{cosx-1+\frac {x^2}{2}-\frac {x^4}{4!}}
=\lim_{x\rightarrow 0}\frac{\frac{x^{3}}{3!}+...+\frac{x^{n}}{n!}}{-\frac{x^{6}}{6!}+...+(-1)^{n}\frac{x^{2n}}{(2n)!}}=\lim_{x\to0}-\frac{6!}{3!\cdot x^{3}}=\infty$

4. Originally Posted by centry57
Take taylor expand:
$e^{x}=1+\frac{x}{1!}+\frac{x^{2}}{2!}+...+\frac{x^ {n}}{n!}+o(x^{n})
cos(x)=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-...+(-1)^{n}\frac{x^{2n}}{(2n)!}+o(x^{2n+1})$

$\lim_{x\to0} \frac {e^x-1-x^2-\frac {x^4}{2}}{cosx-1+\frac {x^2}{2}-\frac {x^4}{4!}}
=\lim_{x\rightarrow 0}\frac{\frac{x^{3}}{3!}+...+\frac{x^{n}}{n!}}{-\frac{x^{6}}{6!}+...+(-1)^{n}\frac{x^{2n}}{(2n)!}}=\lim_{x\to0}-\frac{6!}{3!\cdot x^{3}}=\infty$
Sorry, I made a typo. The first term of the numerator should be $(e^x)^2$instead of $e^x$.
... $=\lim_{x\rightarrow 0}\frac{\frac{x^{6}}{3!}+\frac{x^{8}}{4!}+...+\fra c{x^{2n}}{n!}}{-\frac{x^{6}}{6!}+...+(-1)^{n}\frac{x^{2n}}{(2n)!}}$

I wonder how you got your second last step?

5. $e^{2x}=1+\frac{2x}{1!}+\frac{(2x)^{2}}{2!}+...+\fr ac{(2x)^{n}}{n!}+o((2x)^{n})$
$cos(x)=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-...+(-1)^{n}\frac{x^{2n}}{(2n)!}+o(x^{2n+1})$
$\lim_{x\to0} \frac {e^{2x}-1-x^2-\frac {x^4}{2}}{cosx-1+\frac {x^2}{2}-\frac {x^4}{4!}}$
$=\lim_{x\to0}\frac{1+\frac{2x}{1!}+\frac{(2x)^{2}} {2!}+...+\frac{(2x)^{n}}{n!}-1-x^2-\frac {x^4}{2}}{1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-...+(-1)^{n}\frac{x^{2n}}{(2n)!}-1-x^2-\frac {x^4}{2}}$
$=\lim_{x\to0}\frac{1}{x}\cdot \frac{2+x+\frac{3x^{2}}{4}+\frac{x^{3}}{6}+...+\fr ac{2^{n}(x)^{n-1}}{n!}}{-\frac{3}{2}-\frac{x^{2}}{4}+...+(-1)^{n}\frac{x^{2n-2}}{(2n)!}}$
$=\lim_{x\to0}\frac{1}{x}\cdot \frac{2}{-\frac{3}{2}}=\infty$

6. Anyone can tell me how to edit a formular quickly?
I only use this web :Online LaTeX Equation Editor