Proving limits existance

Question

Im supposed to show and justify if the limit exists of this fucntion. $$\lim_{(x,y)\to(0,0)}\frac{\cos x-1-x^2/2}{x^4+y^4}$$ One way the solution says is fine is to approach along the line $y=0$ which im fine with. so then $$\lim_{x\to0}\frac{\cos x-1-x^2/2}{x^4}$$ But then it says that using l'hopitals rule this is evaluated to $-\infty$ which I do not understand. Im wondering if anyone can explain why this becomes $-\infty$ Thanks in advance!

Answer 1

The first two terms don't cancel. You are left with $\frac{-x^2+\cdots}{x^4}$. The $\cdots$ start with something proportional to $x^4$ so the main thing to worry about is the $\frac{-x^2}{x^4} = \frac{-1}{x^2}$ which is blowing up to $-\infty$.

Alternatively without series:

$$ \frac{\cos x - 1 - x^2/2}{x^4} \to \frac{- \sin x - x}{4x^3}\\ \to \frac{-\cos x - 1}{12x^2} $$

with 2 applications of l'Hopital and then you can show that last quantity is blowing up like $\frac{-2}{12x^2} \to -\infty$.