Question

use the expansion in power of x to calculate the limit

Answer 1

Numerator
\[
N= 1 -e^x = 1 +x -1-x -\frac {x^2}{2!} - \frac {x^3}{3!}- \cdots =-x^2 \left (\frac {1}{2!} + \frac {x}{3!} +\cdots \right)
\]

Answer 2

Denominator
\[
D= 1-\cos(x) = 1 -1 +\frac{x^2}{2!} -\frac{x^4}{4!}+ \cdots = x^2\left(
\frac{1}{2!} -\frac{x^2}{4!}+ \cdots\right)
\]

Answer 3

\[\frac {N}{D} =\frac {-x^2 \left (\frac {1}{2!} + \frac {x}{3!} +\cdots \right)}
{x^2\left(
\frac{1}{2!} -\frac{x^2}{4!}+ \cdots\right)}=\\
\frac {- \left (\frac {1}{2!} + \frac {x}{3!} +\cdots \right)}
{ \left(
\frac{1}{2!} -\frac{x^2}{4!}+ \cdots\right)}-> -1

\]
as x goes to zero.

Answer 4

thank you!

Answer 5

yw