Behavior near a vertical asymptote

As $x \to 0$, the numerator $\cos(x) \to \cos(0) = 1$, a positive constant. The denominator $x^2 \to 0^+$ from both the left and the right, since squaring makes it positive regardless of the sign of $x$.

A positive constant divided by a quantity approaching $0^+$ grows without bound, so both one-sided limits equal $+\infty$: $\lim_{x\to 0^-}\frac{\cos x}{x^2} = +\infty, \quad \lim_{x\to 0^+}\frac{\cos x}{x^2} = +\infty.$ Therefore $\lim_{x\to 0} \frac{\cos x}{x^2} = +\infty$, and since $f$ is undefined at $x=0$ with an infinite one-sided limit there, $x=0$ is a vertical asymptote.

The function grows toward positive infinity. Since cosine of zero equals one, a positive value, and x squared approaches zero from the positive side on both sides of zero, the quotient becomes a large positive number. Both one-sided limits were found to equal positive infinity, so the function increases without bound near x equals zero rather than decreasing.