Question

Help using power series to calculate this limit!

Answer 1

https://gyazo.com/5210c2b2592c1e5f903bbe21c088956e

Answer 2

I under stand that cosx and e^x^2 can be represented as a power series, but I have no idea what to do with the denominator

Answer 3

you need rationalize the denominator
do you know it how ?

Answer 4

use formula a^2 -b^2 = (a-b)(a+b)

Answer 5

still not sure how I would rationalize the denominator. Will you help me out with that? I think i'd know where to go from there

Answer 6

(x→2) (x - 2) / (x² - 4)
= lim (x→0) ((x + 2) - 2) / ((x + 2)² - 4))
= lim (x→0) x / (x(x + 4))
= lim (x→0) 1 / (x + 4)
= 1 / (0 + 4)
= 1/4

Answer 7

Hope it helps!

Answer 8

for the denominator, use a Binomial Expansion
\((1+x^2)^{\frac{1}{4}} - (1-x^2)^{\frac{1}{4}}\)

\(= (1+{\frac{1}{4}} x^2 + \mathcal{O}(x^4)) - (1-{\frac{1}{4}} x^2 + \mathcal{O}(x^4))\)

\(={\frac{1}{2}} x^2 + \mathcal{O}(x^4))\)

Answer 9

yup