Question

Can someone help me to find the following limit?
Thanks in advance

Answer 1

Teach me how to find the sided limites

Answer 2

use l'hôpital's rule

Answer 3

ok let me see I I can do it by the l hopital's rule

Answer 4

I still get something/0

Answer 5

after applying l'hopital's rule

Answer 6

im wrong sorry. Think it worked fine

Answer 7

worked or not?

Answer 8

no. can you show me how to do it, please?

Answer 9

after derivating numerator and denominator simultaneusly and three times I end up with 0/0

Answer 10

ok let me see

Answer 11

\[
\cos(x) \approx 1 -\frac {x^2}2 \\
\text { near } zero
\]

Answer 12

Near zero
\[
1- \cos(x) \approx \frac {x^2} 2\\
\sqrt{1- \cos(x) }\approx \frac x {\sqrt 2}
\]
so
\[
\frac {x} {\sqrt{1- \cos(x) }}\text { is like }\\
\sqrt 2
\]
near zero.
So the limit is
\[\sqrt 2
\]

Answer 13

I used the Taylor Expansions of cos(x) near zero
\[
\cos(x)=\sum_{i=0}^\infty \frac{x^{2n}}{2n!}= 1-\frac {x^2}2 + \frac {x^4} {4!}- \cdots
\]